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\begin{document}

	\begin{center}
	{\huge
		 A non-negative analogue of the Kouchnirenko formula
		}
	\end{center}
\vspace{.1cm}
	\begin{center}
		{\large  Fedor Selyanin}
	\end{center}

		\par\vspace{.1cm}
		\par
	\begin{abstract}
		According to the Kouchnirenko theorem, the Milnor number of an isolated function singularity satisfying certain non-degeneracy condition is equal to an alternating sum (called the Newton number) of the volumes of some polytopes associated with the Newton polyhedron of the singularity. We give a non-negative analogue (without negative summands) of the Kouchnirenko formula and generalize it to a formula for the difference of Milnor numbers of generic singularities with embedded Newton polyhedra. The new formula is supposed to be a hint to the solution of the Arnold's problem on monotonicity of the Newton number. This formula is obtained from the calculation of the asymptotic behavior of critical points of a generic line perturbation of a singularity with a fixed Newton polyhedron, and a new expression for the mixed volume.
	\end{abstract}

	\tableofcontents

	\section{Introduction}
	\graphicspath{{intro/}}
	Recall that the \emph{support} of a polynomial $f\in  \mathbb C[x_1,\dots, x_n]$ is the subset in $\mathbb R^n_{\ge 0}$ consisting of the exponents of monomials with non-zero coefficients in $f(x)$.  Let us denote the support of the polynomial $f$ by $\Fs$. The \emph{Newton polyhedron} $\Fp$ of $f$ is the convex hull of the union of the positive cones with vertices in the points of $\Fs$, $\Fp = Conv(\bigcup\limits_{a \in \Fs} a + \mathbb R^n_{\ge 0})$ (see \cite{sing}). A subset in $\mathbb R^n$ is called a Newton polyhedron if it is the Newton polyhedron of some polynomials.

	Consider a finite set $\Fs\subset \mathbb R^n_{\ge 0}$. Denote by $L_\Fs$ the set of polynomials all exponents of whose monomials are in $\Fs$.
	Consider a  generic polynomial $f\in L_\Fs$ with support $\Fs$ and Newton polyhedron $\Fp$.\label{L_space}

	\begin{theorem}[Kouchnirenko \cite{kush}]\label{th_kush}
			For $\Fp$ \emph{convenient} (which means that the Newton polyhedron $\Fp$ contains a point on each coordinate axis) the Milnor number of $f$ is equal to the the Newton number
			\begin{equation}\label{KF}
			\nu (\Fp) = n! V - (n-1)! \sum\limits_{1\le i \le n} V_i + (n-2)!\sum\limits_{1\le i < j \le n} V_{i,j} - \dots + (-1)^n
			\tag{KF}
			\end{equation}
			of the polyhedron $\Fp$, where $V$ is the $n$-dimensional volume of polytope ``under''\ \ $\Fp$, $V_i$ is the $(n-1)$-dimensional volume of the polytope ``under'' \ \
			$\Fp \cap \{x_i = 0\}$, $V_{i,j}$ is the $(n-2)$-dimensional volume of the polytope ``under''\ \  $\Fp \cap \{x_i = x_j = 0\}$, and so on.
	\end{theorem}

	Recall the Arnold's problem on monotonicity. Consider a convenient Newton polyhedron $\Fp$. The problem consists of two questions:

	\begin{enumerate}
		\item Is there a Newton polyhedron $\Gp\supsetneq \Fp$ such that $\nu(\Gp) = \nu (\Fp)$?
		\item Does there exist the polyhedron $\overline \Fp$ such that $\nu(\overline \Fp) = \nu(\Fp)$ and for every $\Gp \supset \Fp: \nu(\Gp) = \nu(\Fp)$ the polyhedron $\overline \Fp$ contains $\Gp$?
	\end{enumerate}

	The Kouchnirenko Formula (\ref{KF}) implies a formula for the difference between the Milnor numbers of two singularities with embedded Newton polyhedra (one of whose Newton polyhedra is embedded to the other). The corresponding expression is also alternating, and it can hardly be useful to decide whether the two embedded polyhedra have the same Newton number or not.
	In this paper we obtain a non-negative analogue of this expression (see formula (\ref{N-n KF}) on p.\pageref{N-n KF} and theorem \ref{formula} on p.\pageref{formula}). It seems to be a hint to an appropriate answer to the Arnold's problem on monotonicity.

	First we consider two following problems. Having the first problem solved, we obtain a new expression for the difference between the Newton numbers.

	\subsection{Two problems}

	Consider polynomials $f,g \in \mathbb C[x_1,\dots, x_n]$, such that $f(0) = g (0) = 0$. By $\Fs$ and $\Gs$ denote the supports of $f$ and $g$ respectively. By $\Fp, \Gp$ we denote their Newton polyhedra. We assume that $\Fp \subset \Gp$.

	Consider the one-parameter family of functions $f + \e g, \e \in \mathbb C^1$. Let us call it a \emph{line perturbation} of $f$ by $g$. Coordinates of critical points of these functions are expressed by Puiseux series of $\e$. Consider those critical points that are contained in $(\mathbb C\setminus \{0\})^n$ and tend to the origin as $\e \to 0$. Their coordinates correspond to Puiseux series all whose exponents are positive. Suppose $z(\e) = (\alpha_1\e^{a_1} + o(\e^{a_1}), \dots, \alpha_n\e^{a_n} + o(\e^{a_n})), \alpha_i \in \mathbb C\setminus\{0\},$ is such a critical point of $f + \e g$ when $\e\to 0$; then the positive covector $(a_1, \dots ,a_n)$ is called the \emph{asymptotic} of the critical point $z(\e)$. The \emph{multiplicity} of such an asymptotic is the number of critical points with this asymptotic (counted with their multiplicities). The asymptotics and multiplicities are defined in the same way in case of polynomial systems of equations (see problem {\bfseries\rom{2}} below).

	There are some critical points of $f + \e g$ which were not considered above, namely critical points having some identically vanishing coordinates. Here is the formal definition of \emph{dropped} asymptotics.\\

	\begin{definition} \label{lie_def} Let us say that $a_i = +\infty$ if the $i$-th coordinate of the corresponding  curve $z(\e)$ is identically equal to zero.
		An asymptotic $v = (a_1,\dots, a_n)$ is called \emph{dropped} if each $a_i$ belongs to $ \mathbb R_{>0}^* \cup \{+\infty\}$, and some of $a_i$ are equal to $+\infty$ but some are not.
	\end{definition}\\

	We consider two problems in this paper
		\begin{enumerate}
			\item[{\bfseries \rom{1}}]  \textbf{Critical points of a function}

			Let $\Fs,\Gs\subset (\mathbb Z^n_{\ge0} \setminus \{0\})$ be two finite sets. Denote by $\Fp$ and $\Gp$ the corresponding Newton polyhedra $Conv(\bigcup\limits_{a \in \Fs} a + \mathbb R^n_{\ge 0})$ and $Conv(\bigcup\limits_{a \in \Gs} a + \mathbb R^n_{\ge 0})$ and suppose that $\Fp \subset \Gp$. Consider two polynomials $f\in L_\Fs, g\in L_\Gs$ in general position (see definition of $L_\star$ on p.\pageref{L_space}). The goal is to find the asymptotic behavior of the critical points of $f + \e g$ that tend to zero as $\e \to 0$. By finding the asymptotic behavior, we mean listing all asymptotics with their multiplicities assuming that the coefficients of all polynomials ($f$ and $g$ in the problem {\bfseries \rom{1}} and all $f_i$ and $g_i$ in the problem {\bfseries  \rom{2}}) are in a (Zariski) open set in the space of coefficients ($L_\Fs\times L_\Gs$ for the problem {\bfseries \rom{1}} and $L_\Fs^n\times L_\Gs^n$ for the problem {\bfseries \rom{2}}).

			\item[{\bfseries \rom{2}}] \textbf{Roots of a system}

			Consider a generic system of polynomial equations
			$\begin{cases}
			f_1  + \e g_1 = 0 \\
			\dots\\
			f_n + \e g_n = 0
			\end{cases}$, where the Newton polyhedra of all $f_i$ are equal to $\Fp$ and the Newton polyhedra of all $g_i$ are equal to $\Gp$. The goal is to find the asymptotic behavior of all solutions of the system that tend to zero as $\e \to 0$.

		\end{enumerate}

	 In this paper the words ``general position''\ applied to some polynomials or system always mean that collections of their coefficients are in a Zariski open subset of the corresponding set of admissible coefficients.

	We should note that in the problem {\bfseries \rom{1}} the initial data is the supports of the polynomials and not just the corresponding Newton polyhedra. There are many examples of pairs $(\Fs,\Gs), (\Fs_1,\Gs_1)$ such that the Newton polyhedra of $\Fs$ and $\Gs$ coincide with the polyhedra of $\Fs_1$ and $\Gs_1$ respectively but answers to the problem {\bfseries \rom{1}} are different. For example, if $\Gs = \Gs_1 = \{(2,0), (0,2)\}$ and $\Fs = \{(4,0),(0,4)\}, \Fs_1 = \{(4,0),(0,4),(4,1)\}$ then the asymptotics for $(\Fs, \Gs)$ do not coincide with the asymptotics for $(\Fs_1,\Gs_1)$.

	Both problems {\bfseries \rom{1}} and {\bfseries \rom{2}} in this general setting were not considered earlier, but the problem {\bfseries \rom{2}} on roots of a system can be solved by well-known methods (it is even solved in \cite{lin_pert} in the case $\Gs = \{(0,0,\dots,0)\}$ and the solution does not differ in general case much). At the same time the problem {\bfseries \rom{1}} on critical points of a function has a significantly new solution. In this paper we introduce some nontrivial combinatorics to obtain the answer.

	First we solve the problem {\bfseries \rom{2}} on roots of systems, then use this solution as a step in the solution of the problem {\bfseries \rom{1}} on critical points of a function. We will solve also a much more general version of problem {\bfseries \rom{1}} on critical points of a function. The term $\e$ appears in $f + \e g$ only in the power $0$ (at $f$) and in the power $1$ (at $\e g$). It is more convenient to consider any polynomial $h \in \mathbb C [x_1,\dots, x_n,\e]$ instead of $f + \e g$ and to study the asymptotic behavior of corresponding critical point. The obtained problem is called the \emph{generalized problem {\bfseries \rom{1}}}.

	Having the problem {\bfseries \rom{1}} solved, we can sum up all the multiplicities corresponding to all possible asymptotics and obtain a positive analogue of the Kouchnirenko formula (see \S \ref{formula_sec}). We also prove combinatorially that our new formula is identical to the Kouchnirenko formula.

	The same idea of calculating the multiplicities of asymptotics is used in the work \cite{lin_pert}, containing an alternative proof of the Kouchnirenko theorem. But the proof there is a bit tricky, in particular the author neither describes asymptotics of critical points of linear perturbation nor obtains a positive analogue of the Kouchnirenko formula.

	\subsection{Bernstein-Kouchnirenko formula}

	Let $\D_1,\dots, \D_n$ be convex bounded bodies in $\mathbb R^n$, and consider the function

	\begin{equation}
	\chi(\lambda_1,\dots ,\lambda_n)= V(\lambda_1\D_1+\dots +\lambda_n \D_n),\qquad \lambda_{i}\geq 0,
	\end{equation}
	where $V$ stands for the $n$-dimensional volume and its argument is the Minkowski sum of the scaled convex bodies $\D_i$. $\chi$ is a homogeneous polynomial of degree $n$.
	The coefficient of the polynomial $\chi$ at the term $\lambda_1\cdot \lambda_2 \cdots \lambda_n$ multiplied by $n!$ is called the \emph{mixed volume} of $\D_1, \dots, \D_n$ and is denoted by $MV(\D_1,\dots,\D_n)$ (see \cite{book}). For example, for any $\D \subset \mathbb R^n$ the mixed volume $MV(\D,\dots,\D)$ of $n$ copies of it equals $V_N(\D)$, where $V_N$ is the \emph{normalized volume} defined as the usual volume $V$ multiplied by $n!$ (so that for any $\D \subset \mathbb R^n$ we have $V_N(\D) = n! V(\D)$). We use the normalized volumes rather than the usual ones in the paper.

	Consider Laurent polynomials $p_1, \dots, p_n \in \mathbb C[x_1^{\pm1}, \dots, x_n^{\pm1}]$ with supports \mbox{$\mathbf p_1 , \dots \mathbf p_n \subset \mathbb Z^n$} respectively. Denote by $\Delta_i = Conv(\mathbf p_i)$ their Newton polyhedra.

	\begin{theorem} \cite{polyn} \label{kb}
		Suppose the collection of Laurent polynomials $p_1,\dots,p_n$ to be generic in the space of polynomials with given supports. Then the number of solutions $(x_1,\dots,x_n) \in (\mathbb C\setminus 0)^n$ of the system $p_1 = \dots = p_n = 0$ is equal to the mixed volume $MV(\Delta_1, \dots, \Delta_n)$.
	\end{theorem}

	We use a generalized version of this theorem to calculate the multiplicities of asymptotics (see p.\pageref{loc_kush}).

		\subsection{Arnold's problem}

		\begin{enumerate}
			\item \textbf{Original Arnold's problem}

			Let us recall the original Arnold's problem on monotonicity of the Newton number (see \cite{problems} \textnumero 1982-16). Consider a pair of embedded Newton polyhedra $\Fp \subset \Gp$. It is obvious that the Milnor number of a generic singularity with the Newton polyhedron $\Fp$ is not less than the Milnor number of a generic singularity with the Newton polyhedron $\Gp$. So, from the Kouchnirenko theorem we obtain $\nu(\Fp) \ge \nu(\Gp)$. The problem is to find an \emph{elementary} proof of this inequality (\emph{elementary} means that the proof must use the polyhedral geometry only).\label{elem}

			The problem is quite simple in dimensions 2 and 3.

			Indeed, let us consider some two embedded Newton polygons $\Fp \subset \Gp$ of dimension 2. Let us split the difference $\Gp \setminus \Fp$ into some lattice triangles. We can consider the contributions of the triangles to the difference of the Newton numbers separately and then sum them up.

			\begin{figure}[H]
				\center{\includegraphics[scale=0.7]{2d}}
				\caption{\small{Example of splitting. Green points constitute the support $\Fs$ and red points the support~$\Gs$.}}
			\end{figure}

			\begin{ut}
				For each triangle its contribution to the Newton number is non-negative.
			\end{ut}

			\begin{proof}
				Consider a lattice triangle in the positive quadrant. There are two possible cases. If each side of the triangle is not contained in any of the coordinate axes, then its contribution is obviously positive. Otherwise the triangle contains a segment on a single axis as it does not contain the origin. Its height is at least $1$, so the contribution is non-negative.
			\end{proof}

			There are a bit more cases of possible positions for a tetrahedron in case of $n = 3$ but the argument still works.

			\begin{ut}
				For each lattice tetrahedron in $\mathbb Z_{\ge 0}^3$ which can appear in this calculation its contribution to the Newton number is non-negative.
			\end{ut}

			It can be proved by analyzing all possible cases.

			This argument does not work when $n$ is at least $4$. Here is an example of a simplex whose contribution is negative. Consider the tetrahedron $ABCD$ in the coordinate hyperplane $xyz$ with vertices $A(0,0,1), B(0,0,3), C(2,1,1)$ and $D(1,2,1)$ (see picture below). Let us consider also an arbitrary lattice point $E$ in the interior of the positive cone in $\mathbb Z^4$ whose last coordinate equals $1$, for example $E(1,1,1,1)$.

			\begin{ut}
				The contribution of the simplex $ABCDE$ to formula (\ref{KF}) is negative.
			\end{ut}

			\begin{figure}[H]
				\begin{center}
					\includegraphics[scale=0.5]{4d}
				\end{center}
				\caption{simplex $ABCD$}
			\end{figure}
			\begin{proof}

			Its contribution consists of three summands. They correspond to the planes $z$, $xyz$ and $xyzt$, where $t$ is the last coordinate. They are equal respectively to $-2,-6$ and $+6$, the total is hence negative.

			\end{proof}

			A majority of the problems discussed in the paper are non-trivial when $n$ is greater than $3$ only.

			We introduce a positive analogue of the Kouchnirenko formula for the difference between the Milnor numbers of generic singularities with embedded Newton polyhedra. Although the formula is obtained from the solution of the problem {\bfseries \rom{1}} on critical points of a function, it can be considered as an \emph{elementary} proved formula (see p.\pageref{elem}). All the necessary definitions can be formulated in terms of the polyhedral geometry and the equivalence of this positive analogue to the original Kouchnirenko formula is proved on the level of convex geometry (see lemma on p. \pageref{form_lem}). We can formally write a shortened proof of our formula without considering any asymptotics at all which can be considered as a solution of the Arnold's problem. But it is quite unlikely to come up with the positive formula from elementary considerations (without analyzing the multiplicities of asymptotics).

			\item \textbf{Extended Arnold's problem}

			There is another question also known as the Arnold's problem on monotonicity.

			Suppose $\Fp$ to be a Newton polyhedron. The question consists of two parts.
			\begin{enumerate}
				\item How to find out whether there is a lattice expansion $\Fp^\prime \supset\Fp$ of the polyhedron preserving the Newton number? If there are none of them, the polyhedron $\Fp$ is called a \emph{Milnor minimal} polyhedron.
				\item Does the \emph{``Milnor hull''\ $\overline{\Fp}$} of $\Fp$ exists for each $\Fp$? That means whether there exists such a lattice polyhedron $\overline{\Fp}$ that $\nu(\overline{\Fp}) = \nu(\Fp)$ and each lattice polyhedron $\Fp^\prime \supset\Fp, \nu(\Fp^\prime) = \nu(\Fp)$ is embedded in the \emph{``Milnor hull''}\ ($\Fp^\prime \subset\overline{\Fp}$).
			\end{enumerate}

			The problem is solved in case of homogeneous functions (see \cite{hom}). The answer is that it is impossible to expand the Newton polyhedron of a homogeneous isolated singularity without decreasing the Newton number.

			There is an answer to a similar question for surface singularities ($n = 3$) in the paper \cite{3-dim}. Adding a point to the support does not affect the Newton number if and only if the point is contained in some coordinate plane $H$ such that the difference between the ``new''\ and the ``previous''\ polyhedra is a pyramid with base in $H$ and of height equal to 1. The Arnold's problem on monotonicity is defined in a bit different way there.

			We introduce some examples of embedded polyhedra of dimension 4 with the same Newton number in this paper. These examples essentially differ from the ones that appear in case of $n \le 3$.

			I believe that the formula (\ref{N-n KF}) obtained in this paper will help us to solve the Arnold's problem on monotonicity. We are planning to do this in a separate paper.

			\item \textbf{Jumps of the Milnor numbers} \label{jumps}

			The positive analogue of the Kouchnirenko formula should also be useful for the following problem. Let $f: (\mathbb C^n,0) \to (\mathbb C,0)$ be an isolated singularity. Consider the Milnor numbers of all singularities arising in its versal deformation. They form the sequence $\mu_0=\mu(f)>\mu_1>\mu_2> \dots$. The problem is to describe the sequence.

			The set of singularities is a partially ordered sets, namely $g \preceq f$ if $g$ arises in the versal deformation of $f$. In the article \cite{gz} minimal singularities (with respect to the ordering ``$\preceq$'') in two variables among singularities of some fixed modality are discussed. It is proved that if the singularity $f$ is minimal in that sense, then the jump $\mu_0 - \mu_1$ is greater than $1$. So, the first jump is greater than one for example for the singularities $X_9$.

			The problem on the jumps of the Milnor numbers is rather difficult in this general setting. We may consider the Milnor numbers of non-degenerate perturbations only (which means that we fix a support $\Fs \subset (\mathbb Z^n_{\ge 0} \setminus 0)$ and consider all possible supports $\Gs \subset (\mathbb Z^n_{\ge 0} \setminus 0)$ and discuss the Milnor number of the singularities $f + \e g$ at the origin as $\e \to 0$, where $f\in L_{\Fs}, g\in L_{\Gs}$ are in general position). In that case it is enough to discuss the Newton numbers of all polyhedra embedded in $\Fp$.

			The beginnings of the sequence of \emph{non-degenerate jumps of the Milnor numbers} ($\mu_0 - \mu_1$ and $\mu_1 - \mu_2$) are known for singularities of two variables with a convenient Newton polygon consisting of one segment (see \cite{jump_2d2}). The first non-degenerate jump ($\mu_0 - \mu_1$) is also described for Pham singularities in arbitrary dimension (see \cite{pham_jump}).

			Our formula for the difference of the Milnor numbers is useful for calculating some first non-degenerate jumps of the Milnor numbers (see example on p.\pageref{jump_ex}).

		\end{enumerate}

		\subsection{Initial motivation}
		The problem {\bfseries \rom{1}} appears as a subproblem of calculating multiplicities of bifurcation sets (see \cite{mult}) in general case. In the work \cite{mult} the particular case of Pham singularities $f = x_1^{a_1} + x_2 ^{a_2} + \dots + x_n^{a_n}$ and generic linear functions $g$ was considered.

		\subsection{Plan}

		of the arrangement of material in the paper.

		We introduce a step-by step algorithm for calculating the non-negative analogue of the Kouchnirenko formula in \S \ref{algorithm}.

		The problem {\bfseries \rom{2}} on roots of a polynomial system is solved in \S \ref{easy_sec} using dual fans. We also recall the notion of a polynomial's local polytope at an asymptotic and formulate the local version (theorem \ref{loc_kush}) of the Bernstein-Kouchnirenko theorem (theorem \ref{kb}) there.

		In the major part of the paper (\S \ref{n=2_sec} -- \S \ref{check_sec}) we discuss the problem {\bfseries \rom{1}} on critical points of a function. First of all, we demonstrate the answer in case of dimension 2 (\S \ref{n=2_sec}) to introduce the key notions of \emph{internal, semi-internal} and \emph{raised} asymptotics, the concept of \emph{raising} an asymptotic and the \emph{trace polyhedron}. All of them are defined later (in \S \ref{calc_sec}) for arbitrary dimension.

		In \S $\ref{semi_sec}$ we recall the notion of interlaced polytopes (see \cite{polyh}) then define the semi-interlaced polytopes and learn how to find their mixed volumes (theorem \ref{semith}).
		The local polytopes at an internal asymptotics are interlaced (\S \ref{calc_sec}), so the corresponding multiplicities are obtained from theorem \ref{interlaced}.  Then we prove that the local polytopes at a semi-internal asymptotic are semi-interlaced (\S \ref{calc_sec}).

		In \S \ref{calc_sec} we describe the answer to the problem {\bfseries \rom{1}}.
		This answer holds under some genericity assumptions on the leading coefficients of the function; we study these genericity conditions in \S \ref{check_sec}.

		In \S \ref{formula_sec} the positive analogue of the Kouchnirenko formula is obtained by summing up the multiplicities of all asymptotics and simplifying this expression.

		At the end of the paper we give some examples of the asymptotic behavior of critical points of polynomial line perturbations in dimension 3 and 4 (\S \ref{exs_sec}). Examples for dimension 4 are considered to illustrate how the new formula helps to check that some pairs of embedded polyhedra have the same Newton number.

		\subsection{Acknowledgment}

		I am grateful to A.I. Esterov for fruitful discussions and great help with organizing the paper and to V.A. Vassiliev for posing the initial problem (on the multiplicities of bifurcation sets).

		\section{Algorithm for calculating the summands} \label{algorithm}

		\graphicspath{{algorithm/}}

		Let us introduce a step-by-step algorithm for calculating the summands of the non-negative analogue of the Kouchnirenko formula. This algorithm calculates the summands of the formula (\ref{N-n KF}) which appears in the theorem \ref{formula} on p.\pageref{formula}.

		\subsection{Input data}

		The input of the algorithm is a finite subset $\Hs \subset \mathbb Z_{\ge 0}^n \oplus \mathbb Z^1$ such that the linear projection $\pi_\e: \mathbb R_{\ge 0}^n \oplus \mathbb R^1 \to \mathbb R_{\ge 0}^n$ maps $\Hs$ to a set containing a point on each coordinate axis. Suppose $\Hp = \bigcup\limits_{a \in \Hs} a + (\mathbb R_{\ge_0}^n \oplus \mathbb R_{\ge 0}^1)$ to be the Newton polyhedron.

		For instance, let $\Fp \subset \Gp \subset \mathbb R^n_{\ge 0}$ be two Newton polyhedra. Define the polyhedron $\Hp\subset \mathbb R^{n}_{\ge 0} \oplus \mathbb R^1_{\ge 0}$ as $\Hp = Conv (i(\Fp) \cup (i(\Gp) + (0,\dots,0, 1)))$, where $i : \mathbb R^n_{\ge 0} \to \mathbb R^n_{\ge 0} \oplus \mathbb R^1$ is the tautological embedding and $(0,\dots,0, 1) \in \bar 0\oplus\mathbb R^1$ is the unit vector along the $\e$-axis. Then the  non-negative analogue of the Kouchnirenko formula for the difference of Newton numbers $\nu(\Fp) - \nu(\Gp)$ is provided by the output of the algorithm applied to this polyhedron $\Hp$.

		\begin{note}
			The way to choose $\Hp$ to obtain a non-negative analogue of the Kouchnirenko formula is far not the only one. Moving the vertices of $\Hp$ along the $\e$-axis may often not change the corresponding sum (it is a corollary of lemma on p. \pageref{form_lem}). But it may change the summands themselves.
		\end{note}

		In what follows, we consider an arbitrary finite set of points $\Hs: \Hp = \bigcup\limits_{a \in \Hs} a + (\mathbb R_{\ge_0}^n \oplus \mathbb R_{\ge 0}^1)$. It is often convenient to consider $\Hs$ to be the set of all vertices of $\Hp$.

		\subsection{Output data}

		The output is a collection of non-negative (maybe zero) summands. Each summand is an expression containing the normalized volumes of polytopes associated with the set $\Hs$. Having a fixed $\Hp$ and considering different sets $\Hs$ defining this $\Hp$ may change the summands but the total is preserved (it is a corollary of lemma \ref{form_lem} on p.\pageref{form_lem}).

		Namely, the final formula has the form:

		\begin{equation}
		\sum_{v \in IS} \nu(\Tpe(v))m_{v}
		\tag{\ref{N-n KF}}
		\end{equation}
		while the output is the commonwealth of its summands $\{\nu(\Tpe(v))m_{v}|v\in IS\}$ all whose ingredients will be described in the next subsection \ref{inde}.

		\subsection{The indices $\mathbf{v \in IS}$ of summation}
		\label{inde}

		The summands of the desired sum (\ref{N-n KF}) will be indexed by the elements of the set $IS$ which we are going to define now. We need to give a few definitions first.\\

		\begin{definition}\label{sup}
			Consider a polytope $\Ap \in \mathbb R^k$ and a covector $\gamma \in (\mathbb R^k)^*$. As a function on $\Ap$ the linear form $\gamma$ achieves its minimum on a maximal
			compact face of $\Ap$. This face is called the \emph{support face} and denoted by $\Ap^\gamma$.

		The same definition works in case of polyhedron $\Ap$ if the function $\gamma$ is bounded below on the polyhedron $\Ap$.
	\end{definition}\\

		\begin{definition}
			The \emph{dual cone} of a face $K$ of a polyhedron $\Ap \subset \mathbb R^k$ is the set of all such covectors $\gamma\in(\mathbb R^k)^*$ such the support face $\Ap^\gamma$ is $K$.
		\end{definition}\\

		Consider a coordinate plane $E\subseteq \mathbb R^n$ of an arbitrary dimension. Denote by $\mathbb \Hp_E \subset E \oplus \mathbb R^1$ the polyhedron $\Hp \cap (E\oplus \mathbb R^1)$.
		For each bounded facet (a face of codimension 1) $K$ of $\Hp_E$ consider its dual covector of the form $v_K + (0,\dots,0, 1)$, where $v \in E^*_{>0}$ and $(0,\dots,0, 1)$ is the unit covector along the dual of $\e$-axis. Denote by $IS(E)$ the set of all covectors $v_K$ over all bounded facets of $\Hp_E$. Suppose $IS$ to be the union $\bigcup\limits_{E\subseteq \mathbb R^n} IS(E)$ over all the nonempty coordinate subspaces.

		We will see that the set $IS(E)$ splits naturally into the union of subsets $I(E)$ and $S(E)$, where the letters $I$ and $S$ are for ``internal'' and ``semi-internal'' asymptotics (see the definitions on p.\pageref{defsep} and the notations on p.\pageref{notations}).\\

		\textbf{Example}\label{exa}

		Consider support $\Hs$ consisting of the following points\\

		\begin{large}
			\begin{tabular}{ | p{60pt}  c   c  c  c c c|}
				\hline
				Point & $A$ & $B$ & $C$ & $D$ & $E$ & $G_\e$\\[10pt]
				\hline
				Coordinate & $(4,0,0,0)$ & $(0,4,0,0)$ & $(2,0,2,0)$ & $(0,2,2,0)$ & $(0,0,5,0)$ &$(1,0,0,1)$  \\[10pt]
				\hline
			\end{tabular}
		\end{large}\\

		Recall that $\pi_\e$ is the projection $\mathbb R^n_{\ge 0} \oplus \mathbb R^1 \to \mathbb R^n_{\ge 0} $. Denote by $G$ the projected point $\pi_\e(G_\e) = (1,0,0)$.

		\begin{figure}[H]
			\begin{center}
				\includegraphics[scale=0.5]{indices}
			\end{center}
			\caption{picture of projected facets ($IS(\mathbb R^3) = \{(ABCD), (CDEG)\}$)}
		\end{figure}

		The table of facets and corresponding orthogonal covectors is as follows\\

		\begin{large}
			\begin{tabular}{ | p{60pt}  c   c  c  c c c|}
				\hline
				Facet & $ABCD$ & $CDEG$ & $ABG$ & $ACG$ & $CEG$ & $AG$\\[10pt]
				\hline
				Covector & $\frac{1}{3}(1,1,1)$ & $\frac{1}{7}(3,3,2)$ & $\frac{1}{3}(1,1,0)$ & $\frac{1}{3}(1,0,1)$ & $\frac{1}{7}(3,0,2)$ &$\frac{1}{3}(1,0,0)$  \\[10pt]
				\hline
			\end{tabular}
		\label{table}
		\end{large}\\

		\subsection{The multiplicity $\mathbf{m_v}$ of asymptotic $\mathbf{v}$} \label{mv}

		Consider a covector $v\in IS(E) \subset IS$.
		Suppose $K\subset E_{\ge 0} \oplus \mathbb R^1$ to be the facet of $\Hp_E$ corresponding to $v$. Denote by $\Hp_E^{v}$ and $\Hs_E^{v} \subset E$ the polytope $\pi_\e(K)$ and the finite set $\pi_\e(K\cap \Hs)$ respectively.

		Let us consider two cases. The first is a special case of the second one. But it is worth considering them separately since the first one is much simpler. The first case corresponds to \emph{internal} asymptotics and the second to \emph{semi-internal} ones (see p.\pageref{defsep}).

		\begin{itemize}
			\item Suppose there is no coordinate subspace $E^\prime\subsetneq E$ such that $\dim(\Hp_E^{v}\cap E^\prime) = \dim(E^\prime)$.

			Then we set $m_{v}$ to be $V_N(\Hp_E^{v})$ (see corollary \ref{int}).

			\item \label{normalizing1}
			A face $S$ of $\Hp_E^{v}$ is said to be a \emph{suture} (see definition on p.\pageref{semi}) if there exists a coordinate plane $E^\prime \subseteq E$ such that $S \subset E^\prime$ and $\dim (E^\prime) = \dim (S)$. The whole $\Hp_E^{v}$ is always a suture by definition.

			Consider a lattice $L$ in $\mathbb R^k$. We can normalize the volume and the mixed volume for the lattice $L$, namely we set $MV(Q,\dots,Q) = k!$ and $V_N(Q) = k!$, where $Q$ is the primitive cube in $L$.

			Denote by $p_S$ the projection of $\mathbb R^n$ onto the factor $\mathbb R^n / \text{vect$(S)$}$, where vect$(S)$ is the vector space associated with the affine hull aff$(S)$.
			When considering the volume in $S$ it is supposed to be normalized for the lattice $\mathbb Z^n \cap \text{aff$(S)$}$, the volume in $p_S(\mathbb R^n)$ is supposed to be normalized for the image $p_S(\mathbb Z^n)$.

			Suppose $S$ to be a suture of $\Hp_E^{v}$. Let $\mathfrak{v}_S$ be the volume $V_N(S)$. Denote by $\mathfrak{v}$ the vector of the volumes of all sutures of $\Hp_E^{v}$.

			Consider a pair of sutures $S^\prime$ and $S$. If $S^\prime \subset S$ then let us denote by $\zp^S_{S^\prime}$ the closure of $p_{S^\prime}(S) \setminus p_{S^\prime}(Conv((\Hs_E^{v}\cap S) \setminus S^\prime))$. We define numbers $c^{S,S^\prime}$ in the following way

			\begin{enumerate}
				\item $c^{S,S^\prime} = 0$ if $S^\prime$ is not embedded in $S$.
				\item $c^{S,S} = 1$.
				\item For $S^\prime \subset S$ the number $c^{S,S^\prime}$ equals $V_N(\zp^S_{S^\prime})$.
			\end{enumerate}

			Consider the square matrix with entries $c^{S^\prime,S}$, where $S$ and $S^\prime$ run over the set of all sutures of $\Hp_E^{v}$. By conditions 1, 2 above this matrix is upper triangular (in appropriate basis), in particular invertible. Let us denote the inverse of this matrix by $\mathfrak{E}$.

			\label{normalizing2}

			Then we set $m_{v}$ to be the vector's $\mathfrak{E} \cdot \mathfrak v$ component corresponding to the whole polytope $\Hp_E^{v}$ (see theorem \ref{semith}).

		\end{itemize}

		\textbf{Example}

		Let us consider $E$ to be of dimension 3 and the polytope $\Hp_E^{v}$ to be the convex hull of the following points\\

		\begin{large}
			\begin{tabular}{ | p{60pt}  c   c  c  c c|}
				\hline
				Point & $A$ & $B$ & $C$ & $D$ & $G$\\[10pt]
				\hline
				Coordinate & $(4,0,0)$ & $(0,4,0)$ & $(2,0,2)$ & $(0,2,2)$ & $(1,0,0)$  \\[10pt]
				\hline
			\end{tabular}
		\end{large}\\

		\begin{figure}[H]
			\begin{center}
				\includegraphics[scale=0.5]{semi}
			\end{center}
			\caption{example for polytope $\Hp_E^{v}$}
		\end{figure}

		There are 4 sutures. It are the intersections of the polytope with the $xyz$, $xy$, $xz$ and $x$ planes. The vector
		$
		\mathfrak {v} $ is equal to $ \begin{bmatrix}
		36 \\
		12 \\
		6 \\
		3
		\end{bmatrix}
		$. The matrix
		$\mathfrak E $ is equal to $ \begin{bmatrix}
		1 & 2 & 2 & 8 \\
		0 & 1 & 0 & 4 \\
		0 & 0 & 1 & 2 \\
		0 & 0 & 0 & 1
		\end{bmatrix}^{-1} = \begin{bmatrix}
		1 &-2 & -2 & 4 \\
		0 & 1 & 0 & -4 \\
		0 & 0 & 1 & -2 \\
		0 & 0 & 0 & 1
		\end{bmatrix}$.

		For example, the polytope $\zp^{ABCDG}_{AG} = p_{AG}(ABC)$ is shown in Figure \ref{zp}. Its normalized volume is equal to $8$ which is the $(1,4)$ entire of $\mathfrak E^{-1}$.

		\begin{figure}[H]
			\begin{center}
				\includegraphics[scale=1.5]{zp}
			\end{center}
			\caption{\label{zp}  polytope $\zp^{ABCDG}_{AG}$}
		\end{figure}

		$$\mathfrak E \cdot \mathfrak v = \begin{bmatrix}
		1 &-2 & -2 & 4 \\
		0 & 1 & 0 & -4 \\
		0 & 0 & 1 & -2 \\
		0 & 0 & 0 & 1
		\end{bmatrix} \cdot \begin{bmatrix}
		36 \\
		12 \\
		6 \\
		3
		\end{bmatrix} = \begin{bmatrix}
		12 \\
		0 \\
		0 \\
		3
		\end{bmatrix}$$

	 	The desired number $m_{v}$ is therefore equal to $12$. It is equal to the normalized volume of the violet pyramid $BCDG$ in this case.

	 	\begin{figure}[H]
	 		\begin{center}
	 			\includegraphics[scale=.5]{semi_ans}
	 		\end{center}
	 		\caption{pyramid $BCDG$}
	 	\end{figure}

	 	\textbf{Example}

	 	We can continue the table from p.\pageref{table} by adding the values of $m_{v}$

	 	\begin{large}
	 		\begin{tabular}{ | p{60pt}  c   c  c  c c c|}
	 			\hline
	 			Facet & $ABCD$ & $CDEG$ & $ABG$ & $ACG$ & $CEG$ & $AG$\\[10pt]
	 			\hline
	 			Covector & $\frac{1}{3}(1,1,1)$ & $\frac{1}{7}(3,3,2)$ & $\frac{1}{3}(1,1,0)$ & $\frac{1}{3}(1,0,1)$ & $\frac{1}{7}(3,0,2)$ &$\frac{1}{3}(1,0,0)$  \\[10pt]
	 			\hline
	 			$m_v$  & $12$ & $0$ & $0$ & $0$ & $7$ & $3$\\[10pt]
	 			\hline
	 		\end{tabular}
	 	\end{large}\\

		\subsection{The essential polyhedron $\mathbf{\Tpe(v)}$}

		Suppose $v$ to be a covector of $IS(E)$. Covector $v$ is a linear function $v:E\to \mathbb R ^ 1$ by definition. Let us extend this function to the function $\mathbb R^n\oplus \mathbb R^1 = E^\bot \oplus E\oplus \mathbb R^1 \to E^\bot \oplus \mathbb R^1$ mapping $(w^\bot,w,a) \in E^\bot \oplus E\oplus \mathbb R^1$ to $(w^\bot, v(w) + a) \in E^\bot \oplus \mathbb R^1$ and also denote it by $v$.

		Let us denote the point $v(\Hp_E^{v})$ (see the definition of support face on p.\pageref{sup}) by $\mathbf u$ and the Newton polyhedron of the set $v(\Hs)\setminus \mathbf u$ by $\Tp(v)$. Consider the union of faces $\bigcup\limits_u\Tp(v) ^ u$ indexed by such covectors $u\in(E^\bot)^*_{>0} \oplus (\mathbb R^1)^*_{>0}$ that $(v(\Hp))^u = \mathbf u$. The orthogonal projection of this set onto $E^\bot_{\ge 0}$ splits the positive octant into three parts. One of these parts contains the origin, another is the image of this set. The last one is the desired \emph{essential polyhedron} $\Tpe(v)$. This polyhedron may be not convex.\\

		\textbf{Examples}

		\begin{itemize}
			\item Let us give an example of a non-convex essential polyhedron $\Tpe(v)$ first. The support $\Hs$ consists of the following points\\
				\begin{tabular}{ | p{60pt}  c   c  c  c c c c|}
					\hline
					Point & $A$ & $B$ & $C$ & $D$ & $E$ & $F$& $G_\e$\\[10pt]
					\hline
					Coordinate & $(3,0,0,0)$ & $(2,1,0,0)$ & $(2,0,1,0)$ & $(0,1,1,0)$ & $(0,0,5,0)$ &$(0,0,5,0)$ & $(2,0,0,1)$ \\[10pt]
					\hline
				\end{tabular}\\

			Let us denote by $G$ the point $\pi_\e(G_\e) = (2,0,0)$.

			\begin{figure}[H]
				\begin{center}
					\includegraphics[scale=.5]{non-conv}
				\end{center}
				\caption{the Newton polyhedron $\Fp$ and the point $G$}
			\end{figure}

			Let us consider $E$ to be the $x$-axis, covector $v = (1,0,0)$. The essential polytope $\Tpe((1,0,0))$ is as shown in Figure \ref{non-conv}.
			\begin{figure}[H]
				\center{
				\includegraphics[scale=1]{non-conv-es}}
				\caption{\label{non-conv} a non-convex essential polyhedron}
			\end{figure}

			\item
		Let us continue considering the example from p.\pageref{exa}. We are mostly interested in $\Tpe(\frac{1}{3}(1,0,0))$ and $\Tpe(\frac{1}{7}(3,0,2))$, they are as shown in Figure \ref{essential polyhedra}.

		\begin{figure}[H]
			\includegraphics[scale=1]{essential1}
			\includegraphics[scale=1]{essential2}
			\caption{\label{essential polyhedra} the violet polyhedron $\Tpe(\frac{1}{3}(1,0,0))$ and the orange ray $\Tpe(\frac{1}{7}(3,0,2))$}
		\end{figure}

	 \end{itemize}
		\subsection{The Newton number $\mathbf{\nu}$ of a non-convex polyhedron}

		Consider a polyhedron $\Tpe \subset \mathbb R_{\ge 0}^n$ whose complement $ \mathbb R_{\ge 0}^n \setminus \Tpe$ is compact. We define the Newton number $\nu (\Tpe)$ of non-convex polyhedron by the formula (\ref{KF}) in the same way as of the convex ones. And we set $\nu(\Tpe(v))=1$ for $v\in IS (\mathbb R^n)$.

		Every term of the non-negative analogue of the Kouchnirenko formula

		\begin{equation}
		\sum_{v \in IS} \nu(\Tpe(v))m_{v}
		\tag{\ref{N-n KF}}
		\end{equation}

		is therefore defined.\\

		The number $\nu(\Tpe (v))$ is non-negative since it is the sum of non-negative multiplicities of asymptotics \emph{generally raising} from the asymptotic $v$ and the multiplicity of $v$ itself (see definition on p.\pageref{generally raising} and lemma on p.\pageref{form_lem}).\\

		\textbf{Examples}

		\begin{itemize}
			\item The Newton number of the non-convex polyhedron (Figure \ref{non-conv}) is equal to $2-1-1+1 = 1$.

			\item
		We can continue the table on p.\pageref{table}. For example, $\nu (\Tpe(\frac{1}{3}(1,0,0))) = 8-4-2+1 = 3$.

		\begin{large}
			\begin{tabular}{ | p{60pt}  c   c  c  c c c|}
				\hline
				Facet & $ABCD$ & $CDEG$ & $ABG$ & $ACG$ & $CEG$ & $AG$\\[10pt]
				\hline
				Covector & $\frac{1}{3}(1,1,1)$ & $\frac{1}{7}(3,3,2)$ & $\frac{1}{3}(1,1,0)$ & $\frac{1}{3}(1,0,1)$ & $\frac{1}{7}(3,0,2)$ &$\frac{1}{3}(1,0,0)$  \\[10pt]
				\hline
				$m_v$  & $12$ & $0$ & $0$ & $0$ & $7$ & $3$\\[10pt]
				\hline
				$\nu(\Tpe(v))$  & $1$ & $1$ & $1$ & $1$ & $1$ & $3$\\[10pt]
				\hline
			\end{tabular}
		\end{large}\\

		\begin{equation}
		\sum_{v \in IS} \nu(\Tpe(v))m_{v} = 1 \cdot 12 + 1 \cdot 0 + 1 \cdot 0 + 1 \cdot 0 + 1\cdot 7+ 3\cdot 3 = 28
		\end{equation}

		\begin{figure}
			\center{\includegraphics[scale=.5]{Newton}}
			\caption{\label{np}the Newton polyhedron $\Fp$}
		\end{figure}

		And this is the Newton number of the green polyhedron $\Fp$ on Figure \ref{np} because the support $\Hs$ contains the point $G$ whose image under the projection $\pi_\e$ is a vertex of the simplex $x + y + z = 1,\ x,y,z\ge 0$.

	\end{itemize}

		The correctness of this algorithm is provided by the following theorem.
		Suppose a polynomial $h(x_1,\dots,x_n, \e)\in L_{\Hs}$ to be in general position.

		\begingroup
		\def\thetheorem{\ref{formula}}
		\begin{theorem}
	The sum of multiplicities of asymptotics of critical points of polynomial $h$ tending to the origin as $\e \to 0$ is equal to the sum

	\begin{equation}\label{N-n KF}
	\sum_{v \in IS} \nu(\Tpe(v))m_{v}
	\tag{N-n KF}
	\end{equation}
	\end{theorem}
		\addtocounter{theorem}{-1}
		\endgroup

		This theorem is proved on p.\pageref{formula}.

		\section{The problem {\bfseries \rom{2}} on roots of a system} \label{easy_sec}

		\subsection{Formulation of the problem {\bfseries \rom{2}}}

		First of all, let us explain the connection between problems {\bfseries \rom{1}} and {\bfseries \rom{2}}.

		Consider two isolated singularities $f$ and $g$ with embedded Newton polyhedra $\Fp \subset \Gp$ and supports $\Fs$ and $\Gs$ respectively.
		Critical points of the perturbation $f + \e g$ are solutions of the system \mbox{$\forall i\ \frac{\partial (f + \e g)}{\partial x_i} = 0$}. For not dropped asymptotics it is equivalent to the system
		\begin{equation}\label{init}
		\begin{cases}
		x_1 \frac{\partial f}{\partial x_1} +  \e x_1 \frac{\partial g}{\partial x_1} = 0\\
		\dots\\
		x_n \frac{\partial f}{\partial x_n} +  \e x_n \frac{\partial g}{\partial x_n} = 0
		\end{cases}
		\tag{CP}
		\end{equation}
		Denote by $E_i$ the coordinate hyperplane orthogonal to the $i$-th basis vector. Supports of polynomials in the system (\ref{init}) are similar. The support of the $i$-th polynomial is $(\Fs\cup \Gs)\setminus E_i$.

		We will solve the problem {\bfseries \rom{2}} on roots of a system before considering the more complicated problem {\bfseries \rom{1}} on critical points of a function. Let us discuss the asymptotic behavior of solutions of the system
		\begin{equation}\label{easys}
		\begin{cases}
		f_1 + \e g_1 = 0\\
		\dots\\
		f_n + \e g_n = 0
		\end{cases},
		\tag{RS}
		\end{equation}
		where the Newton polyhedra of all polynomials $f_i$ are $\Fp$ and the Newton polyhedra of all polynomials $g_i$ are $\Gp$ and these polynomials are in general position.

		\subsection{A formula for multiplicities of asymptotics}

		In this subsection we describe some tools for calculating the multiplicities of asymptotics.

		Suppose $P(x) = \sum a_k x^k$ to be a polynomial with the Newton polyhedron $\Pp$, where $a_k\ne 0$ and $x^k$ means $x_1^{k_1}\dots x_n^{k_n}$. Consider a polyhedron $\Gamma \subset \mathbb R^n$. By $P(\Gamma)(x)$ denote the polynomial $\sum a_k(\Gamma) x^k$, where $a_k(\Gamma) = 0$ if $k\notin \Gamma$, and $a_k(\Gamma) = a_k$ if $k \in \Gamma$\label{facepol}. By $P^\gamma(x)$ denote the polynomial $P(\mathbb P^\gamma)(x)$. (see definition of the support face on p.\pageref{sup})

		Consider $n$ polynomials $P_1(x,\e), \dots, P_n(x,\e)$. Recall that $\pi_\e: \mathbb R^n \oplus \mathbb R^1 \to \mathbb R^n$ is the projection along the axis corresponding to $\e$. Let $\pi_\e^*: \mathbb C[x,\e] \to \mathbb C[x]$ be the map of plugging $\e = 1$.

		Consider a covector $v$ in $(\mathbb R^n)^*$.\\

		\begin{definition}
			The \emph{local polytope} $\Pp^v$ at the point $v\in(\mathbb R^n)^*$ of a polynomial $P(x,\e) \in \mathbb C[x_1,\dots, x_n,\e]$ is $\pi_\e(\Pp^{v_\e})$, where $v_\e = (v,1) \in (\mathbb R^n)^* \oplus (\mathbb R^1)^*$.
		\end{definition}\\

		\begin{definition}
			\label{nevyr}
			Polynomials $P_1(v,\e), \dots, P_n(v,\e)$ are said to be \emph{generic} if for any covector $\gamma\in (\mathbb R^n)^*$ the system

			\begin{equation}
			(\pi_\e^*(P_1^{v_\e}) )^\gamma = \dots = (\pi_\e^*(P_n^{v_\e})) ^\gamma  = 0
			\end{equation}
			is incompatible in the torus $(\mathbb C\setminus \{0\})^n$.
		\end{definition}\\

		The following theorem is a generalization of the Bernstein-Kouchnirenko theorem (see \cite{polyn}).

		\begin{theorem} \cite{lockal}\label{loc_kush}
			Assume polynomials $P_1(x,\e), \dots, P_n(x,\e)$ to be \emph{generic}.
			Then the multiplicity of asymptotic $v = (v_1, \dots, v_n)$ in the system

			\begin{equation}
			P_1(x,\e) = \dots = P_n(x,\e) = 0
			\end{equation}
			is equal to the mixed volume $MV(\Pp_1^v, \dots, \Pp_n^v)$ of the local polytopes at $v$. Moreover the multiplicity of the solution $(\alpha_1, \dots, \alpha_n)$ of the system

			\begin{equation}
			\pi_\e^*(P_1^{v_\e}) = \dots = \pi_\e^*(P_n^{v_\e}) = 0
			\end{equation}
			is equal to the number of asymptotics of the form  $(\alpha_1 \e^{v_1} + o(\e^{v_1}), \dots, \alpha_n \e^{v_n} + o(\e^{v_n}))$ (counted with their multiplicities).

			Note. Fix some $(\alpha_1 , \dots, \alpha_n) $. There can only be a unique solution of the type $(\alpha_1 \e^{v_1} + o(\e^{v_1}), \dots, \alpha_n \e^{v_n} + o(\e^{v_n}))$ in case of general position (``general position''\ is an abstract condition, whereas system is ``generic''\ if fixed conditions are held).
		\end{theorem}

		\subsection{Dual polytopes}

		Consider a covector $v \in (\mathbb R^n)^*_{>0}$. The multiplicity of the asymptotic $v$ can be described using dual fans.

		We note that all the local polytopes at $v$ of the system (\ref{easys}) are equivalent as all the corresponding Newton polyhedra coincide. So from theorem \ref{loc_kush} we obtain that the multiplicity of $v$ is the normalized volume of the local polytopes at $v$.

		Consider any $w_f\in \Fp^v$ и $ w_g\in \Gp^v$. Denote by $\rho_f(v)$ and $\rho_g(v)$ the expressions $\frac{(w_f,v)}{|v|}$ and $\frac{(w_g,v)}{|v|}$ respectively.\\

		\begin{note}
			If $\rho_f(v) > \rho_g(v)$, then local polytopes at $v$ of polynomials in the system (\ref{easys}) at $v$ are
			\begin{equation}
			\begin{cases}
			\Fp^v & $, if $|v| < \frac{1}{\rho_f(v) - \rho_g(v)}\\
			Conv(\Fp^v, \Gp^v) &$, if $|v| = \frac{1}{\rho_f(v) - \rho_g(v)}\\
			\Gp^v &$, if $ |v| > \frac{1}{\rho_f(v) - \rho_g(v)}
			\end{cases}
			\end{equation}
			If $\rho_f(v) = \rho_g(v)$, then they are equal to $\Fp^v$.
		\end{note}

		An asymptotic of nonzero multiplicity has hence length $|v| = \frac{1}{\rho_f(v) - \rho_g(v)}$ and is unique among the asymptotics with fixed direction. So let us consider the \emph{standard dual simplex}  $\{v = (v_1, \dots, v_n)|v_1 + \dots + v_n = 1, v > 0\}$.

		\begin{sign}
			Denote by $\N (v)$ the asymptotic $\frac{v}{|v|(\rho_f(v) - \rho_g(v))}$.\label{normv}
		\end{sign}

		We split the dual simplex into minimal set of relatively open polyhedral strata for which $\Fp^v$ and $\Gp^v$ are constant in order to describe $\Fp^v$ and $\Gp^v$ as functions of $v$.\\

		\begin{definition}
			\emph{The dual projective fan} $\Fd^*$ of the polyhedron $\Fp$ is the partition of the standard dual simplex to relatively open polyhedral strata $\bigsqcup\limits_{k}\t_k$, where $k$ runs over all cones of the dual fan of polyhedron $\Fp$ and the stratum $\t_k$ is the intersection of the cone $k$ and the standard dual simplex.
		\end{definition}\\

		We can intersect two partitions splitting the same set. \emph{The intersection of partitions} is the partition consisting of all possible intersections of parts of the initial partitions.\\

		\begin{sign}
			Denote by $\Xi$ the intersection of dual projective fans $\Fd^*$ and $\Gd^*$.
		\end{sign}

		\begin{note}\label{cor}
			Consider any two covectors $v_1,v_2$ in $\mathbb (\mathbb R^n)^*_{>0}$. The system
			$
			\begin{cases}
			\Fd^{v_1} = \Fd^{v_2}\\
			\Gd^{v_1} = \Gd^{v_2}
			\end{cases}
			$ holds if and only if the covectors $v_1$ and $v_2$ are contained in the same polyhedral stratum of partition $\Xi$.\\
		\end{note}

		For any stratum $\t\in \Xi$ the faces $\Fp^\t$ and $\Gp^\t$ are well-defined due to Remark \ref{cor}.\\

		\begin{definition}\label{dual_partition}
			The \emph{dual polytope} $\Box^\t$ of a stratum $\t$ is the convex hull $Conv(\Fp^\t, \Gp^\t)$. We also denote by $\Box^v$ this polytope for $v\in \t$.
		\end{definition}

		\subsection{Solution of the problem {\bfseries \rom{2}}}

		The polytope $\Box^\t$ is the local polytope of polynomials in the system (\ref{easys}) at any points $\N(v)$ for $v\in \t$. Taking into account theorem \ref{loc_kush}, we obtain that the multiplicity of any asymptotic $\N(v)$ for $v\in \t$ is equal to the volume $V_N(\Box^\t)$.

		Let us call a polyhedral stratum $\t\in \Xi$ \emph{allowable}\label{allow} if the face $\Fp^\t$ is farther than the face $\Gp^\t$ from the origin (along any direction $v\in \t$).

		\begin{note}\label{dual_dim}
			Suppose a stratum $\t$ to be allowable. Then the sum of dimensions $dim (\t) + dim (\Box^\t)$ is equal to $n$.
		\end{note}

		So asymptotics of the system (\ref{easys}) are $\N(\t_i)$, where $\t_i$ are all allowable zero-dimensional polyhedral strata of the partition $\Xi$.\\

		\begin{note}\label{cover}
			The dual polytopes of the allowable polyhedral strata are splitting the difference $\Gp \setminus \Fp$ into non-intersecting polytopes.
		\end{note}

		From Remarks \ref{dual_dim} and \ref{cover} we obtain the following theorem.

		\begin{theorem}\label{easy}
			All asymptotics of the system (\ref{easys}) are $\N(\t_i)$, where $\t_i$ runs over the set of zero-dimensional polyhedral strata of the partition $\Xi$. Their multiplicities are equal to $V_N(\Box^{\t_i})$ respectively. The sum of the corresponding multiplicities is equal to $V_N(\Gp \setminus \Fp)$.
		\end{theorem}

		\subsection{Dropped asymptotics}

		We also consider asymptotics with some infinite coordinates named \emph{dropped} (see definition on page \pageref{lie_def}). In this subsection we check that the multiplicities of all dropped asymptotics are zero. But there are some dropped asymptotics with non-zero multiplicities in the problem {\bfseries \rom{1}} on critical points of a function.

		Consider an asymptotic $v$ of the form $v_E + v_\infty$, where $v_E \in E^*$ and $v_{\infty}$ is the formal covector with zero coordinates along $E^*$ and infinite coordinates along the orthogonal complement of $E^*$. Note that all monomials whose exponents are not in $E$ are vanishing at the asymptotic $v$. Let us consider the system $(\ref{easys})_E$ obtained from the system $(\ref{easys})$ by crossing out all monomials whose exponents are not in $E$. The system $(\ref{easys})_E$ consists of equations with identical supports and these supports are not empty as the support $\Gs$ contains a point on each coordinate axis. A system of $n$ equations in $dim (E) < n$ variables has no asymptotics in general position. There are hence no dropped asymptotics with nonzero multiplicities.

		\subsection{Examples}

		Let us illustrate the answer with few examples. We do not write coefficients of monomials assuming that it are in general position (the initial coefficients are often not \emph{generic} for some asymptotics). We also assume that $\Fs \subset \Gs$ and do not copy monomials of the polynomial $f$ when defining the polynomial $g$.\\

		\begin{enumerate}
			\item $f = x^7 + x^4y + x^2 y^2 +  x y ^3 + y ^5$

			$g = x ^ 5 +  x y ^2$

			\graphicspath{{easy/}}
			\begin{figure}[H]
				\includegraphics[scale=1.5]{dim2_1}
				\caption{picture of splitting of the difference $\Gp \setminus \Fp$}
			\end{figure}

			There are four 2-dimensional polytopes. Let us show how to find the asymptotic (and its multiplicity) corresponding to the top one (its vertices are $(1,2)$, $(1,3)$ and $(0,5)$). Then we give all the rest answers without details.

			First consider any covector orthogonal to the segment $[(1,3),(0,5)]$, for example, the covector $(2,1)$. Then consider any vector from a point on the face $\Gp^{(2,1)}$ to a point on face $\Fp^{(2,1)}$, for example, the vector $(0,1)$. Now we can calculate that $\N((2,1))$ is equal to $\frac{(2,1)}{((2,1),(0,1))} = (2,1)$. Its multiplicity is equal to the doubled area $2 \cdot$\textonehalf $= 1$ of the corresponding triangle.

			Table of asymptotics and corresponding multiplicities is as follows\\

			\begin{large}
				\begin{tabular}{ | p{150pt}|  c |  c | c | c |}
					\hline
					Asymptotic & $(2,1)$ & $(1,1)$ & $(1,2)$ & $(\frac{1}{2}, \frac{3}{2})$ \\[15pt]
					\hline
					Multiplicity & $1$ & $1$ & $3$ & $2$  \\[15pt]
					\hline
				\end{tabular}
			\end{large}\\

			\item Three-dimensional example:

			$f = x^7y^4z + x^4 y^7z $

			$g = x ^ 6 +  y ^ 6 + x^2y^2 + z^2$

			See the table of points of the supports below (the Newton polyhedron of $f$ does not contain points on each coordinate axis. It is not much important for this problem. It is done so just to simplify the example):

			\begin{large}
				\begin{tabular}{| c  c   c  c    c  c c |}
					\hline
					Point & A & B & C & D  & F & G \\
					\hline
					Coordinate & $(0,0,2)$ & $(0,6,0)$ & $(6,0,0)$ &$(2,2,0)$ & $(7,4,1)$& $(4,7,1)$\\
					\hline
				\end{tabular}
			\end{large}\\

			Illustrations from different point of view are as follows

			\begin{center}
				\begin{figure}[H]
					\includegraphics[scale=.25]{dim3_1}
					\includegraphics[scale=.25]{dim3_2}
					\caption{\label{3-d}three-dimensional example for problem \rom{2}}
				\end{figure}
			\end{center}

			There are three 3-dimensional polytopes on the Figure \ref{3-d}. See the table with asymptotics and corresponding multiplicities below.\\

			\begin{large}
				\begin{tabular}{ | p{150pt}|  c |  c | c |}
					\hline
					Polytope & $ABDG$ & $ACDF$ & $ADFG$ \\[15pt]
					\hline
					Asymptotic & $(\frac{1}{6},\frac{1}{12}, \frac{1}{4})$ & $(\frac{1}{12},\frac{1}{6}, \frac{1}{4})$  & $(\frac{1}{9},\frac{1}{9}, \frac{2}{9})$ \\[15pt]
					\hline
					Multiplicity & $48$ & $48$ & $54$\\[15pt]
					\hline
				\end{tabular}
			\end{large}\\

		\end{enumerate}

		\section{The problem {\bfseries \rom{1}} on critical points of a function, $\mathbf{n = 2}$}\label{n=2_sec}

		The purpose of this section is to give some simple examples of the key notions and constructions of this paper.

		Consider two polynomial singularities $f$ and $g$ in two variables with Newton polyhedra \mbox{$\Fp \subset \Gp$} and supports $\Fs,\Gs$ respectively.
		Let us introduce the asymptotics of line perturbation $f+\e g$ in case of general position.
		We illustrate the answer to the problem \rom{1} in case of dimension $2$ in this section. The answer for arbitrary dimension is given and proved in \S \ref{calc_sec}.

		\subsection{Classification of asymptotics}

		The answers to the problems {\bfseries \rom{1}} and {\bfseries \rom{2}} are quite similar. The following classification is formed by observing the degree of similarity of the answers to the problem {\bfseries \rom{1}} on critical points of a function and to the already discussed (see \S \ref{easy_sec}) problem {\bfseries \rom{2}} on roots of a system.

		\begin{note}
			The partition $\Xi$ splits the interval $((1,0), (0,1))$ into open intervals and points in case of $n = 2$.
		\end{note}

		First let us discuss not dropped asymptotics.\\

		\begin{definition}
			An asymptotic $v\in (\mathbb R_{>0} ^2)^*$ (as well as the corresponding polyhedral stratum $\t \in \Xi: v \in \t$, and its dual polytope $\Box^\t$) is called
			\begin{enumerate}
				\item \emph{Raised} if the polytope $\Box^v$ is contained in a coordinate axis. Else:
				\item \emph{Semi-internal} if the intersection of the polytope $\Box^v$ and a coordinate axis is a line segment.
				\item \emph{Internal}, otherwise.
			\end{enumerate}
		\end{definition}

		\textbf{Examples for the classification}\\

		The supports $\Fs$ and $\Gs$ consist of green and red points respectively. We show the corresponding Newton polyhedra $\Fp, \Gp$ and the splitting of the difference $\Gp \setminus \Fp$ into different classes of polytopes. The orange polytopes are internal, yellow -- semi-internal, violet -- raised ones.\\
		\graphicspath{{dim_2/}}
		\begin{figure}[H]
			\includegraphics[scale = 0.6]{sep_1}
			\includegraphics[scale=0.85]{sep_2}
			\caption{classification of polytopes}
		\end{figure}
		On the left picture the internal polytopes are 4 segments and 3 polygons. There are hence 7 internal polyhedral strata in the partition $\Xi$. Three of them are zero-dimensional and four are one-dimensional. There are also 2 zero-dimensional semi-internal polyhedral strata and 2 one-dimensional raised polyhedral strata.

		As for the right picture, we have 1 zero-dimensional semi-internal polyhedral stratum and 2 one-dimensional picked polyhedral strata.

		Let us discuss the answer for each case in the following subsections \S \ref{int_as}-\ref{rai_as}.

		\subsection{Internal asymptotics}\label{int_as}

		The multiplicity of an internal asymptotic is equal to the doubled volume (i.e. the normalized volume) of its dual polytope $\Box^v$. It is positive if and only if the polyhedral stratum $\t \ni v$ is zero-dimensional ($\t = \{v\}$).

		\subsection{Semi-internal asymptotics}

		Suppose $v$ to be a semi-internal asymptotic. Without loss of generality, assume that the polytope $\Box^v$ contains a segment on the $x$-axis. Denote by $T = (T_x, T_y)$ the closest point to the $x$-axis among all points of $(\Gs^v \cup \Fs ^ v) \setminus Ox$. If the polytope $\Box^v$ does not contain a line segment on the $y$-axis then the multiplicity of asymptotic $v$ is equal to $2 V(\Box_v) - (b-a)\cdot T_y$. We have to subtract a similar to $(b-a)\cdot T_y$ expression for the $y$-axis from the normalized volume $2V(\Box^v)$ additionally otherwise.\\

		\textbf{Example}\\
		\begin{figure}[H]
			\includegraphics[scale=0.85]{semi}
			\caption{example of semi-internal asymptotic}
		\end{figure}

		The multiplicity of $\N(1,1) = (\frac{1}{4},\frac{1}{4})$ is equal to $2V(\Box^v) - (b-a)\cdot T_y = 36 - 4 \cdot 2 = 28$.

		\subsection{Raised asymptotics}\label{rai_as}

		Suppose $v$ to be a raised asymptotic, $v\in \t$. Without loss of generality, assume that the polytope $\Box^v$ is a segment $[a,b]$ on the $x$-axis. Denote by $v_{Ox}$ the covector $(b-a,0)$. Asymptotics in the stratum $\t$ are said to be \emph{raising from} the covector $\frac{v_{Ox}}{|v_{Ox}|}$.

		\emph{The trace polyhedron} $\Tp(\frac{v_{Ox}}{|v_{Ox}|})$ is the Newton polyhedron of the support $((\Gs + v_{Ox}) \cup \Fs) \setminus Ox$.

		Let $v_1, \dots v_k$ be such covectors orthogonal to sides of $\Tp(\frac{v_{Ox}}{|v_{Ox}|})$ that $\forall i : \mathbb X^{v_i} = \{(b,0)\}$, where $\mathbb X = (\Gp + v_{Ox}) \cup \Fp)$. Denote by $m_1, \dots, m_k$ the lengths of projections of the segments $\Tp^{v_1}(\frac{v_{Ox}}{|v_{Ox}|}),\dots, \Tp^{v_k}(\frac{v_{Ox}}{|v_{Ox}|})$ onto the $y$-axis respectively. Then the raised asymptotics are  $\N(v_1), \dots, \N(v_k)$ (see the notation \ref{normv}) and their multiplicities are $m_1\cdot(b-a), \dots , m_k \cdot (b-a)$ respectively.\\

		\textbf{Example}

		The support $\Gs$ is red, $\Fs$ is green. There are the boundaries of Newton polyhedra $\Gp$ and $\Fp$ on the Figure \ref{np-p}. The dual polytope of raised asymptotics is painted violet.

		\begin{figure}[H]
			\center{\includegraphics[scale=0.85]{bas}}
			\caption{\label{np-p}Newton polyhedra}
		\end{figure}

		The next illustration (Figure \ref{es}) shows the corresponding trace polyhedron. We are interested only in those sides of $\Tp(\frac{v_{Ox}}{|v_{Ox}|})$ which do not intersect the punctured gray boundary of the polyhedron $((\Gp + v_{Ox}) \cup \Fp)$.

		\begin{figure}[H]
			\center{\includegraphics[scale=0.85]{move}}
			\caption{\label{es}essential polyhedron (violet)}
		\end{figure}

		Those sides that are orthogonal to the raised asymptotics.

		\begin{figure}[H]
			\center{\includegraphics[scale=0.85]{res}}
			\caption{raised asymptotics}
		\end{figure}

		There are thus two raised asymptotics $\N(v_1) = \N((1,4)) = (\frac{1}{3}, \frac{4}{3})$ and $\N(v_2) = \N((1,3)) = (\frac{1}{3},1)$ in the example. The corresponding multiplicities are $(b-a) \cdot m_1$ and $(b-a) \cdot m_2$ -- both are equal to 3.

		\subsection{Dropped asymptotics}

		An asymptotic is called \emph{dropped} if it has a single infinite coordinate.

		Suppose $v$ to be a dropped asymptotic. Without loss of generality, assume that the $y$-coordinate of $v$ is $+\infty$ (the set of such asymptotics is denoted by $Ox^*_{\infty}$). Consider the vertex $a$ of the ray $\Gp \cap Ox$, the vertex $b$ of the ray $\Fp \cap Ox$ and the point $T = (T_x, T_y)$ -- the closest to the $x$-ray point among all points of the set $(\Gs\cup \Fs) \setminus Ox$. If $a = b$ then there are no dropped asymptotics. If $b>a$ then there is only one dropped asymptotic $(\frac{1}{b-a},+\infty) \in Ox^*_\infty$. Its multiplicity is equal to $(b-a)\cdot (T_y - 1)$.\\

		\textbf{Example}

		\begin{figure}[H]
			\center{\includegraphics[scale=0.85]{lie}}
			\caption{dropped asymptotics}
		\end{figure}

		The multiplicity of dropped asymptotic $(\frac{1}{4},0)$ equals $$(b-a) \cdot (T_y - 1) = 4\cdot (2 - 1) = 4$$.

		\subsection{Example}\label{full2dim}

		Let us consider a brief 2-dimensional example.

		The support of $\Fs$ is as follows

		\begin{center}
			\begin{large}
				\begin{tabular}{| c  c   c  c    c  c |}
					\hline
					Coordinate & $(9,0)$ & $(7,1)$ & $(5,2)$ & $(2,4)$ & $(0,7)$\\
					\hline
				\end{tabular}
			\end{large}\\
		\end{center}

		The polynomial $g$ is a linear function (its support is $\{(1,0), (0,1)\}$).

		The splitting of polytope $\Gp \setminus \Fp$ into the polytopes dual to the polyhedral strata of the partition $\Xi$ is as follows.

		\begin{figure}[H]
			\center{\includegraphics[scale=0.85]{full}}
			\caption{splitting of the polytope $\Gp \setminus \Fp$}
		\end{figure}

		Asymptotics and corresponding multiplicities are as follows

		\begin{center}
			\begin{large}
				\begin{tabular}{| p{120pt}|c | c |  c|  c|}
					\hline
					Asymptotic & $(\frac{1}{3},\frac{1}{6})$ & $(\frac{1}{5},\frac{1}{5})$ & $(\frac{1}{7},\frac{3}{14})$ & $(\frac{1}{8},\frac{1}{4})$\\
					\hline
					Multiplicity & $6$ & $5$ & $14$ & $8$\\
					\hline
				\end{tabular}
			\end{large}\\
		\end{center}
		The first asymptotic is raised, the second and the third are internal, the last is semi-internal.

		The sum of the multiplicities is equal to $33$. That is consistent with the Kouchnirenko formula (\ref{KF}):
		\begin{equation}
		2S - l_x - l_y + 1 = 48 - 9 - 7 + 1 = 33
		\end{equation}

		\section{Interlaced and semi-interlaced polytopes}\label{semi_sec}
		\graphicspath{{semi/}}

		\subsection{Interlaced polytopes}

		The problem {\bfseries \rom{1}} on critical points of a function is reformulated as a problem of finding the asymptotic behavior of roots of a polynomial system (see the system (\ref{init}) on p.\pageref{init} for not dropped asymptotics). This is why we can use theorem \ref{loc_kush} to calculate the corresponding multiplicities. We give a more exact answer to the problem {\bfseries \rom{1}} in this paper.

		To obtain a more precise answer we need to calculate some mixed volumes. We can use the notion of \emph{interlaced polytopes} and theorem \ref{interlaced} to do this.

		Consider polytopes $\D_1, \dots, \D_n$. Denote by $U_{\D_1,\dots,\D_n}$
		the convex hull of the union
		$\bigcup_i \D_i$.\\

		\begin{definition}
			The polytopes $\D_1,..,\D_n$ are said to be \emph{interlaced} if for any covector $\gamma \in (\mathbb R^n)^*\setminus\{0\}$ there exists $I\subset \{1,\dots,n\}, |I| = dim (U_{\D_1,\dots,\D_n} ^ v) + 1$ such that $\D_i^v \subset U_{\D_1,\dots,\D_n} ^ v$ for any $i\in I$.
		\end{definition}\\

		\begin{theorem}\cite{polyh}\label{interlaced}
			If the polytopes $\D_1,..,\D_n$ are interlaced then their mixed volume \\$MV(\D_1,\dots, \D_n)$ is equal to $V_N(U_{\D_1,..,\D_n})$.
		\end{theorem}

		We use theorem \ref{loc_kush} and calculate the mixed volume of local polytopes to obtain the multiplicities of asymptotics.
		While calculating the multiplicities in the problem {\bfseries \rom{1}} on critical points of a functions, we face more complicated (than interlaced) sets of polytopes. In this paper we generalize theorem $\ref{interlaced}$ to the case when the estimation $|I| = dim (U_{\D_1,..,\D_n} ^ \gamma) + 1$ is replaced by $|I| = dim (U_{\D_1,..,\D_n} ^ \gamma)$ (we also have to add some new assumptions) in the following subsections \ref{semi-def}--\ref{semi-calc}. This type of polytopes appear when considering semi-internal asymptotics in the problem \rom{1}. We say that these polytopes are \emph{semi-interlaced} and their mixed volume can be calculated due to theorem \ref{semith} (p.\pageref{semith}). This is a new formula for the mixed volume which is valuable in itself.

		\subsection{Definition of semi-interlaced polytopes}\label{semi-def}

		Suppose $\As\subset \mathbb Z^n$ to be a finite set of lattice points and denote by $\Ap$ its convex hull $Conv(\As)$.\\

		\begin{definition}
			Consider a polytope $\Dp$ with vertices in the set $\As$. Consider all faces of the polytope $\Ap$ which do not meet the sub-polytope $\Dp$. Denote by $\Df$ the set of thrown out faces consisting of all maximal (inclusion-wise) faces among them. The polytope $\Dp$ is said to be a \emph{daughter sub-polytope} of the pair of polytope and support $(\Ap, \As)$ if the following conditions are held
			\begin{enumerate}
				\item The set $\Df$ consists of pairwise disjoint faces.
				\item Polytope $\Dp$ is the convex hull $Conv(\As \setminus \bigcup\limits_{K\in \Df} K)$.
			\end{enumerate}
		\end{definition}

		Informally speaking, a \emph{daughter sub-polytope} is the convex hull of its support which is obtained from the set $\As$ by throwing out all points which are contained in some set of disjoint faces. But it must be done carefully so that no other face is accidentally thrown out (that may happen if, for example, there is a $1$-dimensional face without any interior points of support $\As$ in it. We throw out both its ends as $0$-dimensional faces and consider the segment not to be thrown out. This situation is prohibited by definition). \\

		\begin{definition}\label{semi}
			Daughter sub-polytopes $\Dp_1, \dots , \Dp_n$ of the pair $(\Ap, \As)$ are said to be \emph{semi-interlaced} in $(\Ap, \As)$ if for any covector $\gamma \in (\mathbb R^n)^* \setminus \{0\}$  the set $I = \{i|\Dp_i^\gamma \subset \Ap ^ \gamma\}$ contains at least $dim(\Ap ^ \gamma)$ elements. If there are exactly $dim(\Ap ^ \gamma)$ elements, then the face $\Ap ^ \gamma$ is called a \emph{suture} of $(\Ap,\As)$.
		\end{definition}\\

		This definition is similar to the definition of interlaced polytopes (see p.\pageref{interlaced}). But the condition of being daughter sub-polytope is quite significant. For example, if we do not require sub-polytopes to be daughter sub-polytopes, then each pair of polygons on plane would be semi-interlaced in their convex hull, whereas the square and the rhombus on Figure \ref{rh_sq} below are not semi-interlaced in any pair $(\Ap,\As)$. It is even impossible to translate one of those polygons to make them be semi-interlaced in a pair $(\Ap, \As)$.

		\begin{figure}[H]
			\center{\includegraphics[scale=0.85]{non_int}}
			\caption{\label{rh_sq}example of not semi-interlaced polytopes}
		\end{figure}

		\begin{ut}
			Suppose sutures $S_1$ and $S_2$ to be intersecting and $\text{aff}(S_1\cap S_2) = \text{aff} (S_1) \cap \text{aff}(S_2)$, where ``aff'' is the affine hull. Then their intersection $S_1 \cap S_2$ and their ``union''\ $\Ap \cap \text {aff} (S_1 \cup S_2)$ are also sutures.
		\end{ut}

		\begin{proof}
			Let face $S$ be the intersection of these sutures $S_1\cap S_2$. Denote by $d_1, d_2$ and $d$ the dimensions of faces $S_1, S_2$ and $S$ respectively. Let $g$ be the number of polytopes (among the semi-interlaced polytopes $\Dp_1,\dots,\Dp_n$) which neither meet $S_1$ nor $S_2$. Denote by $g_1$ the number of semi-interlaced polytopes which do not meet $S_1$ but do meet $S_2$. The number $g_2$ is defined similarly.

			Then the following system holds
			\begin{equation}\label{eq}
			\begin{cases}
			g + g_1 = n - d_1\\
			g + g_2 = n - d_2,
			\end{cases}
			\end{equation}

			as the faces $S_1$ and $S_2$ are sutures. From the system $(\ref{eq})$ we obtain

			\begin{equation}\label{ll}
			g_1 +g _2 + g = 2n - d_1 - d_2 - g
			\end{equation}

			From the definition of daughter sub-polytopes we obtain that each of $g$ polytopes not meeting both faces $S_1$ and $S_2$ do not meet the face $\Ap \cap \text{aff}(S_1\cup S_2)$ neither. Its dimension is $d_1 + d_2 - d$ due to the assumption $\text{aff}(S_1\cap S_2) = \text{aff} (S_1) \cap \text{aff}(S_2)$. The definition of semi-interlaced polytopes for the faces $S$ and $\Ap \cap \text {aff} (S_1 \cup S_2)$ claims that

			\begin{equation}
			\begin{cases}\label{ineq}
			g + g_1  + g_2 \le n - d\\
			g  \le n - d_1 - d_2 + d
			\end{cases}
			\end{equation}

			Taking into account the equality $(\ref{ll})$ we obtain that both the inequalities in system $(\ref{ineq})$ have to be equalities and both faces $S_1\cap S_2$ and $\Ap \cap \text {aff} (S_1 \cup S_2)$ are sutures.

		\end{proof}

		\subsection{Calculating mixed volume of semi-interlaced polytopes}\label{semi-calc}

		Let the polytopes $\Dp_1, \dots , \Dp_n$ be semi-interlaced in the pair $(\Ap, \As)$ and $\Df_1, \dots ,\Df_n$ be the sets of thrown out faces respectively. Our goal is to calculate their mixed volume $MV(\Dp_1, \dots , \Dp_n)$.

		\begin{note}\label{semi_ind}
			Suppose $S = \Ap^v$ to be a suture of the pair $(\Ap,\As)$. Then the polytopes  $\{\Dp_i^v| \Dp_i^v \subset S\}$ are semi-interlaced in $(S, S\cap \As)$.
		\end{note}

		We are using the same notations as in \S \ref{mv}. The only new notion is the following one.
		For each suture $S$ denote by $\tilde{\mathfrak{v}}_S$ the mixed volume of the polytopes $\{\Dp_i^v| \Dp_i^v \subset S\}$ in aff$(S)$. Denote by $\tilde{\mathfrak{v}}$ the vector of the corresponding mixed volumes.

		\begin{note}
			Suppose each face of $\Ap$ to be a suture of $(\Ap, \As)$. Then the entires of $\mathfrak{E}$ are the Euler obstructions of a toric variety. The obstructions are defined in \cite{euler_def} and the expression  similar to the formula (\ref{S-I}) for the mixed volume of semi-interlaced polytopes see in \cite{euler}.
		\end{note}

	\begin{theorem}\label{semith}
		Suppose $\As\subset \mathbb Z^N$ to be a finite set, polytope $\Ap$ to be the convex hull $Conv(\As)$ and the polytopes $\Dp_1, \dots , \Dp_n$ to be semi-interlaced in the pair $(\Ap, \Ap)$. Then the following equality holds
		\begin{equation}\label{S-I}
		\tilde{\mathfrak{v}} = \mathfrak{E} \cdot \mathfrak{v}
		\tag{S-I}
		\end{equation}
	\end{theorem}

	\begin{proof}
		Suppose polynomials $f_1,\dots, f_n \in L_{\As}$ (see definition of $L_\star$ on p.\pageref{L_space}) to be in general position. Denote by $f_i^\e$ the polynomial obtained from $f_i$ by multiplying by $\e$ all its monomials whose exponents contained in any face of the set $\Df_i$.\\

		\begin{note}
			\begin{enumerate}
				\item The local polytope of the polynomial $f_i^\e$ at the covector $0\in(\mathbb R^n)^*$ is $\Dp_i$.
				\item The sum of multiplicities of all the system's \begin{equation}\label{zpp}
				\begin{cases}
				f_1^\e = 0\\
				...\\
				f_n^\e = 0
				\end{cases}
				\end{equation} asymptotics is equal to $V_N(\Ap)$.\\
			\end{enumerate}
		\end{note}

		The first part of the remark above is obvious, the second is a special case of lemma 2 on p.15 of the paper \cite{hova}.

		\renewcommand{\O}{\mathbf O}
		Thus, $V_N(\Ap) - MV(\Dp_1, \dots \Dp_n)$ is the sum of multiplicities of system's (\ref{zpp}) nonzero asymptotics. Consider an asymptotic $v\in(\mathbb R^n)^*\setminus \{0\}$. Due to theorem \ref{loc_kush}, its multiplicity equals the mixed volume of the local polytopes at $v$. If $\Dp_i^v \subset \Ap^v$, then the $i$-th polynomial's of the system (\ref{zpp}) local polytope at $v$ is $\Dp_i^v$. Suppose the face $\Ap^v$ not to be a suture. Then there are at least $dim (\Ap^v) + 1$ of local polytopes contained in a face of dimension $dim (\Ap^v)$, so the mixed volume is zero. This means that the asymptotic $v$ may be of nonzero multiplicity only if the face $\Ap^v$ is a suture.

		Fix a suture $S$. Let us calculate the sum of multiplicities of such asymptotics $v$ that $\Ap^v = S$. Denote by $\Dp_i(v)$ the $i$-th local polytope at the asymptotic $v$. The set $I = \{i|\Dp_i \subset S\}$ is, by definition, of $dim (S)$ elements and the mixed volume of the polytopes $MV\{\Dp_i|i\in I\}$ (in aff$(S)$) is $\tilde{\mathfrak{v}}_S$. Then the mixed volume of the local polytopes at the asymptotic $v$ equals $\tilde{\mathfrak{v}}_S$ multiplied by the mixed volume $ MV\{p_S(\Dp_i)|i\notin I\}$ in $p_S(\mathbb R^n)$. Let us denote the latter mixed volume by $m(v)$.

		Denote by $\As_S$ the set $p_S(\As) \cap \zp_S^\Ap$ (see definition of $\zp_S^\Ap$ in \S \ref{mv}) and by $\O_S$ the point $p_S(S)$. Our goal is to prove that the sum of multiplicities of all asymptotics $v:\Ap^v = S$ is equal to $V_N(\zp_S^\Ap)$. We need to prove the following lemma first.

		\begin{lemma}\label{only}
			Suppose $\Dp$ to be a daughter sub-polytope of the pair $(\Ap,\As)$ and $\mathcal D$ to be the corresponding set of thrown out faces. Denote by $W\in \mathcal D$ the face of the polytope $\Ap$ which contains the suture $S$. Then for each face $W^\prime \in \mathcal D \setminus \{W\}$ its projection $p_S(W^\prime)$ does not meet the set $\As_S$.
		\end{lemma}

		\begin{proof}
			Assume the converse.

			Suppose $X\in W^\prime$ to be such a point that $p_S(X)\in\As_S$. Let $U$ be the minimal face of polytope $\Ap$ containing the point $X$. Denote by $U_S$ the projection $p_S(U)$ of the polytope $U$.

			If the polytope $U_S$ contains the point $\O_S$ then the face $U$ intersects the face $S$. Then $W^\prime \equiv W$ due to paragraph 1 of daughter sub-polytope definition since $U\subset W^\prime$ and $S \subset W$.

			If $\O_S\notin U_S$, then the polytope $U_S$ is contained in a common face of the polytopes $\zp_S^\Ap$ and $Conv(\As_S\setminus\O_S)$. Then there exists such covector $\gamma_S \in (\mathbb R^n / \text {vect}(S))^* \setminus \{0\}$ that the following conditions are applied:

			\begin{itemize}
				\item $p_S(\Ap)^{\gamma_S} = \O_S$
				\item $\left[ p_S(\As) \setminus \zp_S^\Ap\right]^{\gamma_S}\supset U_S$
				\item $\gamma_S(U_S) - \gamma_S(\O_S) = 1$.
			\end{itemize}

			Note that a covector in $(\mathbb R^n / \text {vect}(S))^*$ can be extended to a covector in $(\mathbb R^n)^*$.

			Consider a covector $\gamma_U \in (\mathbb R^n)^*$ such that the support face $\Ap^{\gamma_U}$ is $U$. Consider the covector $\gamma = \gamma_U + (\gamma_U(S^{\gamma_U}) - \gamma_U(U)) \gamma_S$. Then the face $\Ap^\gamma$ intersects both faces $S$ and $U$ but not the polytope $\Dp$. Then $W \equiv W^\prime$ since $S\subset W$ and $U \subset W^\prime$.

		\end{proof}

			Let us illustrate the proof of lemma \ref{only} above with an example.
			Suppose that the support $\As$ consists of the following points\\

			\begin{large}
				\begin{tabular}{| c  c   c  c    c  c c |}
					\hline
					Point & A & B & C & D  & E & X \\
					\hline
					Coordinate & $(0,0,0)$ & $(0,0,3)$ & $(4,0,0)$ &$(2,2,2)$ & $(0,6,0)$& $(3,1,1)$\\
					\hline
				\end{tabular}
			\end{large}\\

			Let the suture $S$ be the edge $AB$. We assume that the point $X$ belongs to a face $W^\prime$, where $W^\prime \in \mathcal D\setminus \{W\}$.\\

			\begin{figure}[H]
				\center{\includegraphics[scale=.8]{lem3d}}
				\caption{\label{lem3d}the polytope $\Ap$, the support $\As$ and the suture $S$ (blue)}
			\end{figure}

			\begin{figure}[H]
				\center{\includegraphics[scale=2]{lem2d}}
				\caption{\label{lem2d}the polytope $p_S(\Ap)$, the polytope $\zp_S^\Ap$ (blue) and the point $\O_S$ in $\mathbb R^3 / S$}
			\end{figure}

			The face $U_S$ is the segment $[p_S(C), p_S(B)]$, the face $U$ is the segment $[CD]$ and the polytope $\zp_S^\Ap$ is the non-convex quadrangle $(\O_S,p_S(B), p_S(C), p_S(D))$ in this example.

			There is the unique covector $\gamma_S = \frac{1}{4}(1,1)$ satisfying corresponding conditions in this case. We can consider $\gamma_U = (-1,0,-1)$. Then the face $S^{\gamma_U}$ is the point $B$ and the covector $\gamma$ is equal to $\gamma_U + (-3 - (-4))\gamma_S = \frac{1}{4}(-3,1,-4)$. The support face $\Ap^\gamma = CDB$ does not intersect the polytope $\Dp$ but intersects both faces $S$ and $U$.

			\begin{lemma}\label{zpl}
				The sum of the mixed volumes $m(v)$ over all covectors $v: \Ap ^v = S$ equals $V_N(\zp_S^\Ap)$.
			\end{lemma}

		\begin{proof}
			Consider a polynomial of $x_1,\dots, x_n$ whose support is $\As$. Let us multiply all its monomials whose exponents are in the suture $S$ by $\e$ and denote the obtained polynomial by $f_S^\e$. Let $\D(v)$ be the image of the local polytope of $f_S^\e$ at $v$ under the projection $p_S$.

			The polytope $\D(v)$ is obviously $\O_S$ or $Conv(\As_S^v)$ or $Conv(\O_S, \As_S^v)$.

			\begin{ut}
				Let us consider 3 cases:
				\begin{enumerate}
					\item If $\D(v) = \O_S$, then the mixed volume of the local polytopes $MV\{p_S(\Dp_i(v))\}, i \notin I$ equals zero.
					\item If $\D(v) = Conv(\O_S,\As_S^v)$, then the local polytopes $\{p_S(\Dp_i(v))\}, i \notin I$ are interlaced and their convex hull is $\D(v)$.
					\item If $\D(v) = Conv(\As_S^v)$, then the mixed volume of the local polytopes $\{p_S(\Dp_i(v))\}, i \notin I$ equals zero.
				\end{enumerate}
			\end{ut}

			\begin{proof}
				\begin{enumerate}
					\item If $\D(v) = \O_S$, then all $p_S(\Dp_i), i \notin I$ are also $\O_S$ and the mixed volume of points is equal to zero.

					\item For each $i\notin I$ consider the face $W_i\supset S, W_i\in \Df_i$. Taking into account lemma \ref{only} we obtain that $p_S(\Dp_i(v)) = Conv(\As_S^v \setminus p_S(W_i), \O_S)$.

					Consider a face $K$ of the polytope $\D(v)$. Let us prove that at least $dim(K) + 1$ projections of the local polytopes meet the face $K$.

					If $\O_S \in K$, then every polytope $p_S(\Dp_i(v)), i \notin I$ contains the point $K$ and the condition is applied.

					If $\O_S\notin K$, then we will show by contradiction that there are at least $dim(K) + 1$ projections of the local polytopes meeting the face $K$. Assume there are $n - dim(S) - dim(K)$ polytopes $\Dp_i(v), i\notin I$ such that $p_S(\Dp_i(v)) \cap K = \emptyset$. Denote by $U$ the minimal face of the polytope $\Ap$ containing the suture $S$ such that $K \subset p_S(U)$. Its dimension is not less then $dim(K) + dim(S) + 1$. But the power of the set $\{j\notin I| \exists U^\prime\in \Df_j : U \subset U^\prime\}$ is at least $n - dim(S) - dim(K)$. Thus, the power of the set $\{j\notin I| \nexists U^\prime\in \Df_j : U \subset U^\prime\}$ is at most  $dim(S) + dim(K)$. This contradicts the definition of semi-interlaced polytopes.

					\item Let us prove that there are $dim(Conv(\As_S^v)) + 1$ projections of the local polytopes contained in  $Conv(\As_S^v)$. This obviously implies the mixed volume to be zero.

					Consider the normalized covector $cv$ instead of $v$ such that $\Delta(cv) = Conv(\As_S^v, \O_S)$. Note that $c > 1$. So we have obtained the previous case and there are \mbox{$dim(Conv(\As_S^v)) + 1$} projections of the local polytopes meeting the face $Conv(\As_S^v)$. Note that if $p_S(\Dp_i(cv))$ meets the face $Conv(\As_S^v)$, then the polytope $p_S(\Dp_i(v))$ is $p_S(\Dp_i(cv)) \cap Conv(\As_S^v)$. Thus, $p_S(\Dp_i(v))$ is contained in the face $ Conv(\As_S^v)$. So we obtained that there are $dim(Conv(\As_S^v)) + 1$ projections of the local polytopes contained in  $Conv(\As_S^v)$.

				\end{enumerate}
			\end{proof}

			Taking into account theorem \ref{interlaced}, we obtain that the summed up mixed volumes of local polytopes over all asymptotics $v:\Ap^v = S$ is equal to the sum of normalized volumes of $\D(v)$ over all $v : \D(v) = Conv(\O_S,\As_S^v)$. Note that for all others $v$ the volume $V_N(\D(v))$ is $0$. These polytopes split $\zp_S^\Ap$ into disjoint parts. So the sum of their normalized volumes equals $V_N(\zp_S^\Ap)$.
		\end{proof}
			From lemma \ref{zpl} we obtain the equality
			\begin{equation}\label{S_IR}
			MV(\Dp_1, \dots , \Dp_n) = V_N(\Ap) - \sum\limits_{S  \in \mathcal S} V_N(\zp_S^\Ap) \cdot  \tilde{\mathfrak{v}}_S
			\tag{S-IR}
			\end{equation}
		Formulas (\ref{S_IR}) and (\ref{S-I}) are obviously equivalent.
	\end{proof}

	\subsection{Example of semi-interlaced polytopes}

	We give an example of two polytopes semi-interlaced in a rectangle. The support $\As$ consists of all marked points, the polygon $\Ap = ABCD$ is the green rectangle (see Figure \ref{s-e}). The daughter polygons are obtained by throwing out pairs of opposite sides of this rectangle ($AD$ and $BC$ for the red one $\Dp_1$, $AB$ and $CD$ for the blue one $\Dp_2$).

	\begin{figure}[H]
		\includegraphics[scale=2]{exs}
		\caption{\label{s-e}semi-interlaced polytopes}
	\end{figure}

	Every 0-dimensional and 1-dimensional face is a suture in this example. So the mixed volume of the blue and the red polygons equals the following expression

	\begin{equation}\label{smth}
	\begin{matrix}
	V_N(ABCD) -  V_N(\zp^\Ap_A) \cdot \tilde{\mathfrak{v}}_A -  V_N(\zp^\Ap_B) \cdot \tilde{\mathfrak{v}}_B-  V_N(\zp^\Ap_C) \cdot \tilde{\mathfrak{v}}_C-  V_N(\zp^\Ap_D) \cdot  \tilde{\mathfrak{v}}_D - \\ - V_N(\zp^\Ap_{AB}) \cdot \tilde{\mathfrak{v}}_{AB} -  V_N(\zp^\Ap_{BC}) \cdot  \tilde{\mathfrak{v}}_{BC}-  V_N(\zp^\Ap_{CD}) \cdot  \tilde{\mathfrak{v}}_{CD}-  V_N(\zp^\Ap_{DA}) \cdot  \tilde{\mathfrak{v}}_{DA}
	\end{matrix}
	\end{equation}

	Let us calculate each term separately.

	\begin{enumerate}
		\item $\tilde{\mathfrak{v}}_A = \tilde{\mathfrak{v}}_B = \tilde{\mathfrak{v}}_C = \tilde{\mathfrak{v}}_D = 1$.
		\item The polytope $\zp^\Ap_A$ looks as follows

			\begin{figure}[H]
				\begin{center}
				\includegraphics[scale=1.5]{A_out}
				\caption{the polytope $\zp^\Ap_A$}
			\end{center}
			\end{figure}

		Thus, $V_N(\zp^\Ap_A) = V_N(\zp^\Ap_B) = V_N(\zp^\Ap_C) = V_N(\zp^\Ap_D) = 5$.

		\item Consider the suture $AD$.
		\begin{figure}[H]
			\begin{center}
				\includegraphics[scale=1.5]{AD_on}
				\caption{the suture $AD$}
			\end{center}
		\end{figure}
		Both its endpoints $A$ and $D$ are sutures, $\zp^{AD}_A = AJ$ and $ \zp^{AD}_D = DJ$. So, $\tilde{\mathfrak{v}}_{AD} = l(AD) - l(AJ) - l(DJ) = 4-2-2 = 0$. That was obvious from the beginning as the volume of the point $J$ on the line equals $0$.

		\item $\tilde{\mathfrak{v}}_{AB} = l(AB) - l(AE) - l(BF) = l(EF) = 1$.
		\begin{figure}[H]
			\begin{center}
				\includegraphics[scale=1.5]{AB_on}
				\caption{the suture $AB$}
			\end{center}
		\end{figure}

		\item $V_N(\zp^\Ap_{AB}) = 1$.
		See the illustration of the segment $\zp^\Ap_{AB}$ below (Figure \ref{last}).

		\begin{figure}[H]
			\begin{center}
				\center{\includegraphics[scale = 4]{AB_out}}
				\caption{\label{last}the polytope $\zp^\Ap_{AB}$}
			\end{center}
		\end{figure}
		\item Now we plug everything in formula (\ref{smth}) and obtain the mixed volume of the blue and the red polygons

		\begin{equation}
		2 \cdot 7 \cdot 4 - 4 \cdot 5\cdot 1 - 2\cdot 1 \cdot 1 - 2 \cdot 0 = 34
		\end{equation}

	\end{enumerate}

	\section{Solution of the generalized problem {\bfseries \rom{1}} on critical points of a function} \label{calc_sec}

	In this section we generalize the problem {\bfseries \rom{1}} on critical points and solve the obtained generalized problem. Let us say a few words about what the generalization is done for.

	We can classify the polyhedral strata for arbitrary $n$ similarly as in case $n = 2$. We are going to use the induction hypothesis for smaller $n$ to obtain the multiplicities of raised asymptotics. But the problem that appears for smaller $n$ is more general, so we have to generalize the induction hypothesis to obtain the answer for the initial problem {\bfseries \rom{1}}.

	\subsection{Generalized problem {\bfseries \rom{2}} on roots of a system}

	The initial problems {\bfseries \rom{1}} and {\bfseries \rom{2}} concern polynomials of the form $f + \e g$, where $f,g \in \mathbb C^n[x_1,\dots, x_n]$ are polynomials with embedded Newton polyhedra $\Fp \subset \Gp$. It is convenient to generalize these problems and consider polynomials $h\in \mathbb C^n[x_1,\dots,x_n][\e]$ instead of $f + \e g$. Suppose $\Hs$ and $\Hp$ to be the support and the Newton polyhedron (in $\mathbb R_{\ge0}^n\oplus \mathbb R^1$) of $h$ respectively. The projection $\pi_\e (\Hp)$ (along the $\e$ axis) is assumed to contain a point on each coordinate axis.

	It is worth solving the generalized problem {\bfseries \rom{2}} on roots of a polynomial system before considering the generalized problem {\bfseries \rom{1}} on critical points of a function. So, we are interested in asymptotics of the system

	\begin{equation}\label{easy_gen}
	\begin{cases}
	h_1 = 0\\
	\dots\\
	h_n = 0
	\end{cases},
	\tag{G-RS}
	\end{equation}
	where these polynomials $h_i$ are in general position and have the prescribed Newton polyhedron $\Hp$. We should consider the \emph{dual projective fan} $\Hd^*$ instead of the partition $\Xi$.\\

	\begin{definition} \label{veer}
		\emph{The dual projective fan $\Hd^*$} of the polyhedron $\Hp$ is the set of intersections of dual fan's of $\Hp$ relatively open cones  with the hyperplane $(x,1), x \in \mathbb R^n$.
	\end{definition}\\

	Theorem $\ref{easy}$ can be obviously generalized to the following statement

	\begin{theorem}
		Asymptotics of the system (\ref{easy_gen}) are such covectors $v$ that $dim(\Hd^v) = n$ (or, equivalently, the polyhedral stratum $\t \in \Hd^* : v \in \t$ is zero-dimensional). Their multiplicities are equal to corresponding numbers $V_N(\Hp^v)$.
	\end{theorem}

	\subsection{Stating the generalized problem {\bfseries \rom{1}}}

	\begin{definition}
		Consider a support $\Hs \in \mathbb R^n_{\ge 0} \oplus \mathbb R^1$.
		Consider a polynomial $h \in L_\Hs$ and assume it to be in general position. \emph{The generalized problem {\bfseries \rom{1}} on critical points of a function} is the problem on finding asymptotics of those critical points of $h$ that tend to zero as $\e \to 0$.
	\end{definition}\\

	Recall that in the initial problem {\bfseries \rom{1}} on critical points of a function we explored asymptotics of line perturbation $f + \e g$. When considering not dropped asymptotics it is equivalent to finding asymptotics of the system

	\begin{equation}
	\begin{cases}
	x_1 \frac{\partial f}{\partial x_1} +  \e x_1 \frac{\partial g}{\partial x_1} = 0\\
	\dots\\
	x_n \frac{\partial f}{\partial x_n} +  \e x_n \frac{\partial g}{\partial x_n} = 0
	\end{cases}
	\tag{\ref{init}}
	\end{equation}

	Similarly, not dropped asymptotics of critical point of the generalized problem {\bfseries \rom{1}} are asymptotics of the system

	\begin{equation}\label{gen}
	\begin{cases}
	x_1 \frac{\partial h}{\partial x_1} = 0\\
	\dots\\
	x_n \frac{\partial h}{\partial x_n} = 0
	\end{cases}
	\tag{G-CP}
	\end{equation}

	Dropped asymptotics are discussed separately in \S \ref{lie_subsec}. As for now, we assume the polynomials of the system (\ref{gen}) to be generic if the polynomial $h\in L_\Hs$ is in general position (see definition on p.\pageref{nevyr}). This is proved in the next section (\S \ref{check_sec}).

	In the problem {\bfseries \rom{2}} on roots of a system the only asymptotics proportional to $v$ which may be of nonzero multiplicity is $\N(v)$. In the problem {\bfseries \rom{1}} on critical points of a function this also holds true. This follows from the following statement about the generalized problem {\bfseries \rom{1}}.

	\begin{ut} \label{maxi}
		Suppose an asymptotic $v$ to have a positive multiplicity. Then there exists a coordinate plane (of an arbitrary dimension) $E\subset \mathbb R^n$ such that $\Hp^v\subset E$ and $dim(E) = dim (\Hp^v)$.
	\end{ut}
	\begin{proof}
		Assume the converse. Then there exists an asymptotic of positive multiplicity such that the minimal coordinate plane $E \supset \Hp^v$ satisfies $dim(E) > dim (\Hp^v)$. There are hence $dim(E)$ local polytopes (at $v$) contained in $\Hp^v$, but $dim(\Hp^v) < dim(E)$. Thus, the mixed volume of the local polytopes is equal to zero.
	\end{proof}

	We describe asymptotics by induction on the dimension $n$.

	\subsection{Classification of polyhedral strata}

	We split the polyhedral strata of the dual projective fan $\Hd^*$ (see the definition on p.\pageref{veer}) into 3 types. The classification is based on the degree of similarity of the answer to the problem {\bfseries \rom{1}} on critical points of a function and the problem {\bfseries \rom{2}} on roots of a system. For example, the answers are the same for asymptotics belonging to \emph{internal} polyhedral strata and differ dramatically for the asymptotics belonging to \emph{raised} polyhedral strata.\\

	\begin{definition} \label{defsep}
		A polyhedral stratum $\t \in \Hd^*$ (as well as every asymptotic $v\in \t$ and its dual polytope $\Hp^\t$) is said to be
		\begin{enumerate}
			\item \emph{Raised} if $\Hp^\t$ is contained in some proper coordinate subspace $E\subsetneq \mathbb R^n$. Else:
			\item \emph{Semi-internal} if there exists a coordinate plane $E \subsetneq \mathbb R^n$ such that $dim(\Hp^\t\cap E) = dim(E)$.
			\item \emph{Internal} otherwise.
		\end{enumerate}
	\end{definition}

	Dropped asymptotics are considered separately in \S \ref{lie_subsec}.

	Let us discuss the simplest cases of internal and semi-internal asymptotics first.

	\subsection{Internal and semi-internal polyhedral strata}

	Internal asymptotics and their multiplicities coincide with the ones found in the generalized problem {\bfseries \rom{2}} on the roots of a polynomial system. Semi-internal asymptotics are the same but their multiplicities differ.

	Let us prove a lemma first. Denote by $E_i$ the coordinate hyperplane $\{x_i = 0\}$.

	\begin{lemma}\label{int_semi}
		Suppose $\mathbf A \subset \mathbb Z_{\ge0}^n$ to be a finite subset of the lattice which is not contained in any of the hyperplanes $E_i$. Consider the polytopes $\{\D_i\}: \D_i =  Conv(\mathbf A \setminus E_i)$. Then the given polytopes $\D_1,\dots,\D_n$ are semi-interlaced in the pair $(Conv(\mathbf A), \mathbf A)$. The sutures are such faces $K$ of $Conv(\mathbf A)$ that there exists a coordinate plane $E(K) \subset \mathbb R^n :  K\subset E(K)$ and $dim (E(K)) = dim (K)$.
	\end{lemma}
	\begin{proof}
		The polytopes $\D_i$ are obviously daughter sub-polytopes of $(Conv(\mathbf A), \mathbf A)$ (thrown out faces are $\Df_i = \{K_i\} = \{Conv(\mathbf A) \cap E_i\}$).

		Consider a face $K$ of the polytope $Conv(\mathbf A)$. Let $E$ be the minimal coordinate plane containing $K$. The number of polytopes $\D_i$ meeting the face $K$ equals $dim (E)$. Thus, the polytopes are semi-interlaced in $(Conv(\mathbf A), \mathbf A)$ and the face $K$ is a suture if and only if there is a coordinate space of dimension $dim(K)$ containing $K$.
	\end{proof}

	\begin{ut}\label{ok}
		Suppose $\t$ to be an internal or semi-internal polyhedral stratum. Consider any $v \in \t$ and denote by $\D_1(v),\dots, \D_n(v)$ the local polytopes at the point  $v$. Then $\D_i(v) = Conv((\Hs^\t)\setminus E_i)$ and the polytope $\Hp^\t$ is not contained in any coordinate hyperplane.
	\end{ut}

	By definition of internal and semi-internal polyhedral strata the polytope $\Hp^\t$ is not contained in any coordinate hyperplane. The local polytopes are the mentions above ones, because the support of the $i$-th equation of the system (\ref{gen}) is $\Hs \setminus E_i$. Using lemma \ref{int_semi} we obtain the following conjectures:

	\begin{conj}\label{int}
		Suppose $\t$ to be an internal polyhedral stratum, $v\in\t$. Then the polytopes $\D_1(v), \dots, \D_n(v)$ are interlaced and their convex hull is the polytope $\Hp^\t$. Thus, due to theorem \ref{interlaced}, an internal asymptotic has nonzero multiplicity if and only if it belongs to a zero-dimensional polyhedral stratum. Its multiplicity equals $V_N(\Hp^\t)$.
	\end{conj}
	\begin{conj}\label{semi-int_cor}
		A semi-internal asymptotic may have nonzero multiplicity only if it belongs to a zero-dimensional polyhedral stratum.
	\end{conj}

	\begin{note}
		Let $\t = \{v\}$ be a zero-dimensional semi-internal polyhedral stratum. Due to lemma \ref{int_semi}, the polytopes $\D_1(v), \dots, \D_n(v)$ are semi-interlaced in the pair $(\Hp^\t,\Hs^\t)$. The sutures are such faces $K$ that there exists a coordinate plane $E(K) : K \subset E(K)$ and $dim(K) = dim (E(K))$.

		The multiplicity of asymptotic $v$ can be therefore calculated using theorem \ref{semith}.
	\end{note}

	\subsection{Raised asymptotics}\label{pick_subsec}

	Consider a coordinate subspace $E \oplus \mathbb R^1 \subsetneq \mathbb R ^ n \oplus \mathbb R^1$. Let us restrict the support $h$ to that subspace and obtain $\Hs_E = \Hs \cap (E \oplus \mathbb R^1)$. Denote by $h_E$ the function $h(E \oplus \mathbb R^1)$ (see the notation on p.\pageref{facepol}) and by $\Hp_E$ its Newton polyhedron in $E$. Let us consider the generalized problem {\bfseries \rom{1}} on critical point of $dim (E)$ variables for the support $\Hs_E$.

	\begin{sign}\label{intersys}
		Denote by (\ref{gen})$_E$ the system describing not dropped asymptotics of this problem.
	\end{sign}

	We assume asymptotics for this problem to be known by induction hypothesis.

	Let us introduce another notation.

	\begin{sign}\label{subsys}
		Denote by (\ref{gen})($E$) the subsystem of system (\ref{gen}) consisting of all lines corresponding to the basis of $E$.
	\end{sign}

	\begin{definition}
		An asymptotic $v \in (\mathbb R^n)^*_{>0}$ is said to be \emph{raising from} an asymptotic $v_E^{(i)} \in (E)^*_{>0}$ if
		\begin{enumerate}
			\item $\Hp^v = \Hp_E^{v_E^{(i)}}$.
			\item The polytope $\Hp^v$ is not contained in any proper coordinate subspace of $E$.
		\end{enumerate}
	\end{definition}

	\begin{note}
		Every raised asymptotic $v$ defines the unique asymptotic $v_E^{(i)} \in \partial \mathbb R^n_{\ge 0}$ from which it is raising. The asymptotic $v_E^{(i)}$ is an internal or semi-internal asymptotic of the system (\ref{gen})$_E$. Consider a coordinate plane $E$. Every internal and semi-internal asymptotic $v_E^{(i)} \in E$ of the system (\ref{gen})$_E$ defines a nonempty polyhedral stratum $\t(v_E^{(i)}) \in \Hd^*$\label{pick_poly} of asymptotics raising from $v_E^{(i)}$. The stratum $\t(v_E^{(i)})$ is the dual of the face $\Hp_E^{v_E^{(i)}}\in E$ polyhedral stratum.
	\end{note}

	From the claim \ref{maxi} we obtain that a raising from $v_E^{(i)}$ asymptotic may have positive multiplicity only if $dim(\Hp_E^{v_E^{(i)}}) = dim(E)$.

	Consider an asymptotic $v_E^{(i)} \in E$. Suppose $m_{v_E^{(i)}}$ to be its multiplicity in the system (\ref{gen})$_E$. Our goal is to describe the multiplicities of asymptotics raising from $v_E^{(i)}$, or equivalently belonging to $\t(v_E^{(i)})$.

	Denote by $E^\bot$ the orthogonal complement of $E$ of $\mathbb R^n$.

	\begin{note}
		The local polytopes of the system  (\ref{gen})($E$) at $v$ coincide with the local polytopes of system (\ref{gen})$_E$ at $v_E^{(i)}$. The mixed volume of the local polytopes of the system (\ref{gen}) at $v$ is hence equal to $m_{v_E^{(i)}}$ multiplied by the mixed volume of projections of the local polytopes of system (\ref{gen})($E^\bot$) onto $E^\bot$.
	\end{note}

	Thus, we should consider projections of the local polytopes of the system (\ref{gen})($E^\bot$) onto $E^\bot$. Let us introduce a construction describing the projections of the local polytopes and calculating their mixed volume by induction. The crucial notion here is the \emph{trace polyhedron} which is about to be defined.

	\begin{sign}
		An asymptotic $v_E^{(i)}$ is a covector and it defines the projection $\mathbb R^{dim(E)} \oplus \mathbb R^1 \to \mathbb R^1$. Let us also denote by $v_E^{(i)}$ the linear projection $\mathbb R^n\oplus\mathbb R^1 \to \mathbb R^{n-dim(E)}\oplus\mathbb R^1$ which acts on $\mathbb R^{dim(E)} \oplus \mathbb R^1$ as mentioned earlier and which is identical on $E^\bot$.
	\end{sign}

	The image of the lattice $\mathbb Z^n \oplus \mathbb Z^1$ under projection $v_E^{(i)}$ contains the standard lattice $\mathbb Z^{n-dim(E)} \oplus \mathbb Z^1$ and can be obtained from it by multiplying the last coordinate by some $\frac{1}{\xi}, \xi \in \mathbb N$. Let us normalize $\mathbb R^{n-dim(E)}\oplus \mathbb R^1$ for the image of the standard lattice. Thus, the image of the standard lattice is the standard lattice in $\mathbb R^{n-dim(E)}\oplus \mathbb R^1$.\\

 	\begin{definition}
 		\emph{The trace polyhedron} $\Tp(v_E^{(i)})$ is the Newton polyhedron of the support $v_E^{(i)}(\Hs \setminus (E\oplus \mathbb R^1))$.
 	\end{definition}\\

 	Denote $\mathbf E_v = v_E^{(i)}(\Hs \setminus E) \in \mathbb R^{(n-dim(E))} \oplus \mathbb R^1$. By induction hypothesis, asymptotics and the multiplicities are assumed to be known in the $(n-dim(E))$-dimensional problem {\bfseries \rom{1}} with the support $\mathbf E_v$. Denote the obtained system for not dropped asymptotics by (\ref{gen}$-\mathbf E_v$). Denote this system's asymptotics by $u_j$ and their multiplicities by $m^{u_j}$ respectively.

 	\begin{theorem}
 		\begin{enumerate}
 			\item The polyhedral stratum $\t(v_E^{(i)})$ is the set of asymptotics $v_E^{(i)} + \xi u$ such that $u\in E^\bot$ and $(v_E^{(i)}(\Hp))^u = \{0\}$.
 			\item The multiplicity of asymptotic $v = v_E^{(i)} + \xi u_j$ equals $m^{u_j} m_{v_E^{(i)}}$. All the other asymptotics are of zero multiplicity.
 		\end{enumerate}
 	\end{theorem}

	\begin{proof}
		The first part is obvious, the second is a corollary of the following statement:
		\begin{ut}
			Suppose $(v_E^{(i)}(\Hp))^u = \{0\}$. Then the projections of the local polytopes of the system (\ref{gen})($E^\bot$) at the point $v_E^{(i)} + \xi u$ onto $E^\bot$ coincide with the local polytopes of the system (\ref{gen}$-\mathbf E_v$) at the point $u$.
		\end{ut}

		Recall that the $i$-th local polytope of the system (\ref{gen}) is $\D_i (v_E^{(i)} +\xi u) = [Conv(\As\setminus E_i)]^{v_E^{(i)} + \xi u}$. It is obvious that $v_E^{(i)}\left([Conv(\As\setminus E_i)]^{v_E^{(i)} + \xi u} \right) = \left(Conv( \mathbf E_v\setminus [E_i \cap E])\right)^{u}$, because all we change is the order of commuting operations. The claim is therefore proved.
	\end{proof}

	All asymptotics (except for the dropped ones) and their multiplicities are thus described.

	\subsection{Dropped asymptotics} \label{lie_subsec}

	Let us discuss dropped asymptotics.
	Denote by $E^*_\infty$ the space of covectors $v_E^{(i)} + v_E^\infty$ such that $v_E^{(i)} \in (E)^*_{>0}$ and $v_E^\infty$ has zero coordinates along $(E)^*$ and $+\infty$ along $(E^\bot)^*$. Recall that asymptotic $v$ is said to be dropped if it belongs to some $E^*_\infty$, where $E$ is an arbitrary proper coordinate subspace. We describe the multiplicities of dropped asymptotics in this section in case of general position.

	Suppose $v \in E_\infty$ to be a dropped asymptotic. Then all monomials whose exponents do not belong to $E$ do vanish. Asymptotics and the multiplicities of the subsystem $(\ref{init})(E)$ (see notation on p.\pageref{subsys}) become the same as for the system (\ref{gen})$_E$ (see p.\pageref{intersys}).

	\begin{theorem} \label{lie_th}
		The multiplicity of asymptotic $v = v_E^{(i)} + v_E^\infty \in E^*_\infty$ equals the multiplicity of asymptotic $v_E^{(i)}$ in the system (\ref{gen})$_E$ multiplied by the Milnor number of a generic singularity of $n - dim(E)$ variables with support $p_E(\Hs) \setminus \{0\}$, where $p_E$ is the orthogonal projection $\mathbb R^n \oplus \mathbb R^1 \to E^\bot$.
	\end{theorem}
		\begin{proof}
		Denote by $i_1,\dots, i_k$ the numbers of basis vector spanning the subspace $E^\bot$.
		Fix a solution curve of the system (\ref{gen})$_E$. Its multiplicity in the system (\ref{gen}) is equal to its multiplicity in (\ref{gen})$_E$ multiplied by the multiplicity of zero solution of system
		\begin{equation}
		\begin{cases}
		\frac{\partial }{\partial x_{i_1}} (h)= 0\\
		\dots \\
		\frac{\partial }{\partial x_{i_k}} (h)= 0
		\end{cases}
		\end{equation}
		in which the fixed solution of the system (\ref{gen})$_E$ is plugged.
		We must not multiply equations by $x_{i_j}$ as $x_{i_j} = 0$. The multiplicity of zero solution of this system is the Milnor number of a generic singularity with the support $p_E(\Hs)\setminus \{0\}$.
		Thus, the multiplicity of each asymptotic of the system (\ref{gen})$_E$ is multiplied by the Milnor number of a generic singularity with the support $ p_E(\Hs) \setminus \{0\}$.
	\end{proof}
	\begin{conj}\label{no_lie}
		If the set $p_E(\Hs)$ contains a vertex of the simplex $\begin{cases}
		x_1 + \dots + x_{n - dim(E)} = 1\\
		\forall i: x_i \ge 0 \\
		\end{cases}$, then there are no dropped asymptotic $v \in E^*_\infty$.\\
	\end{conj}

	\begin{note}
		The Milnor number is the sum of asymptotics multiplicities of the singularity at a linear morsification (see the example on p.\pageref{full2dim}). We may assume that the support $\widetilde{\Gs}$ consists of all vertices of the simplex $\begin{cases}
		x_1 + \dots + x_{n - dim(E)} = 1\\
		\forall i: x_i \ge 0 \\
		\end{cases}$ and that the support $\widetilde{\Fs}$ is the support of the considered singularity. Thus, the problem of finding the Milnor number is contained in the problem {\bfseries \rom{1}} on critical points of a function and it is assumed to be solved by induction. There are also no dropped asymptotics due to corollary \ref{no_lie}.
	\end{note}

	\subsection{Remarks}

	We prove a couple of trivial statements to make it easier to understand the overall picture in this subsection. The results of this subsection are not important neither for further calculations nor for proves.

	Assume that we know the Newton polyhedron $\Hp$ but do not fix the support. Let us discuss what can we say having such incomplete data.

	\begin{ut}
		Consider an internal or semi-internal asymptotic $v_E^{(i)}$ of the system (\ref{gen})$_E$ (see notation \ref{intersys}). Denote by $\mathbf V_E^{(i)}$ the set of all semi-internal asymptotics $v$ (of the initial system) such that $\Hp_E^{v_E^{(i)}} \subset \Hp^v$. Then all asymptotics raising from $v_E^{(i)}$ are contained in the Newton polyhedron of the support $\mathbf V_E^{(i)} \subset v_E^{(i)} \oplus (E^\bot)^*$.
		All the dropped asymptotics are contained in the infinite points of those Newton polyhedra ($u$ is said to be an infinite point of this polyhedron if there exists such a point $u_f$ in the polyhedron, that $u$ is obtained from $u_f$ by changing some coordinates corresponding to $E^\bot$ to infinity).
	\end{ut}

\begin{proof}
	It is obvious since all the vertices of $\t(v_E^{(i)})$ (see notation  in Remark \ref{pick_poly}) are zero-dimensional polyhedral strata. So, the union of vertices of $\t(v_E^{(i)})$ is equal to $\mathbf V_E^{(i)}$ and the polyhedral stratum $\t(v_E^{(i)})$ is the Newton polyhedron of $\mathbf V_E^{(i)}$.
\end{proof}

	\begin{ut}
		Assume that for every semi-internal $n$-dimensional polytope $\Delta$ and for every proper coordinate subspace $E \subsetneq \mathbb R^n:dim(E\cap\Delta) = dim(E)$, it is known that $p_E(\Delta)$ (see the notation of $p_E$ in theorem \ref{lie_th}) is the simplex $\sum x_i \le 1, x_i \ge 0$. Then there are only internal and semi-internal asymptotics and the answer to the generalized problem \rom{1} does not depend on the support $\Hs$ (but on the Newton polyhedron $\Hp$ of $\Hs$ only).
	\end{ut}

	\begin{proof}
		The formula of the multiplicity of dropped asymptotics contains the multiplier equal to the Milnor number of some generic singularity. The singularity contains a linear summand due to our assumption, and its Milnor number is $0$. Thus, there are no dropped asymptotics.

		Consider an internal or semi-internal asymptotic $v_E^{(i)}$ of system the (\ref{gen})$_E$. The difference between the projection of the trace polyhedron $p_E(\Tp (v_E^{(i)}))$ and $p_E(\Hp)$ is the simplex $\sum x_i \le 1, x_i \ge 0$. So there are no raising from $v_E^{(i)}$ asymptotics.

		There are therefore only internal and semi-internal asymptotics and they do depend on the polyhedron $\Hp$ only.
	\end{proof}

	\section{Checking genericity and simplicity}\label{check_sec}

	In this section we finally prove that the system (\ref{gen}) is generic (see definition on p.~\pageref{nevyr}) in case of general position of $h\in L_\Hs$ . We also prove that every not dropped asymptotic is simple.\label{symple} An asymptotic $a = (a_1,\dots,a_n)$ is called \emph{simple} if for any two solutions $(\alpha_1 \e^{a_1} + o(\e^{a_1}), \dots, \alpha_n \e^{a_n} + o(\e^{a_n}))$ and $(\beta_1 \e^{a_1} + o(\e^{a_1}), \dots, \beta_n \e^{a_n} + o(\e^{a_n}))$ with the asymptotic $a$ the covectors of leading coefficients $(\alpha_1,\dots, \alpha_n)$ and $(\beta_1,\dots, \beta_n)$ are different (and there are no multiple solutions).

	It is not necessary to check genericity of dropped asymptotics as we calculated their multiplicities without using theorem \ref{loc_kush} (it is used only in the induction hypothesis).

	\subsection{Genericity}

	Let us first prove the following lemma.

	\begin{lemma} \label{kush_check}
		Suppose a support $\As \subset \mathbb Z^n_{\ge 0}$ to be contained in some hyperplane but not in any of the coordinate hyperplanes. Then a polynomial $p \in L_\As$ does not have critical points in the torus $(\mathbb C\setminus\{0\})^n$ in general position.
	\end{lemma}

	\begin{proof}
		Assume that $\As$ is contained in a hyperplane orthogonal to covector $\gamma = (\gamma_1,\dots, \gamma_n)$. Then $\gamma_1 x_1 \frac{\partial p}{\partial x_1} + \dots + \gamma_n x_n \frac{\partial p}{\partial x_n} = c p$, where $c\ne 0$ as the hyperplane containing $\As$ does not pass through the origin. Thus, the equality $dp = 0$ implies $p = 0$.

		If  $\mathbf x \in (\mathbb C\setminus\{0\})^n$ is a critical point of the polynomial $p$, then the following system holds

		\begin{equation}
		\begin{cases}\label{part1}
		dp(\mathbf x) = 0\\
		p(\mathbf x) = 0
		\end{cases}
		\end{equation}

		Note that we can multiply the polynomial $p$ in the system (\ref{part1}) by any monomial $q$ of $x_i$ and $x_i^{-1}$ so that the obtained system is also applied
		\begin{equation}\forall q:
		\begin{cases}
		d(pq)(\mathbf x) = 0\\
		pq(\mathbf x) = 0
		\end{cases}
		\end{equation}

		We can therefore assume that the support $\As$ contains the origin and the polynomial $p$ has a nontrivial free member. The polynomial $p$ defines the map $\mathbb C^n \to \mathbb C$. We obtain from the Sard's lemma that the set of critical values has measure $0$, so for almost all free members the system (\ref{part1}) has no solutions.
	\end{proof}

	Denote by $h_i$ the polynomial $x_i \frac{\partial}{\partial x_i} h$.

	Let us prove that for every asymptotic $v\in (\mathbb R^n)^*_{>0}$ and for every covector \mbox{$\gamma \in (\mathbb R^n)^*\setminus \{0\}$} the system (\ref{gen}) is generic (see definition on p.\pageref{nevyr}) in general position by induction on $n$. We check that the following system has no solutions in the torus $(\mathbb C\setminus\{0\})^n$

	\begin{equation}\label{check_sys}
	\begin{cases}
	\pi_\e^*(h_1^{v_\e}) ^\gamma = 0\\
	\dots\\
	\pi_\e^*(h_n^{v_\e}) ^\gamma  = 0
	\end{cases},
	\tag{$*$}
	\end{equation}

	where $v_\e = (v,1) \in (\mathbb R^n)^* \oplus (\mathbb R^1)^*$.

	Suppose that $\Hp^v \subset E_{>0}$, where $E$ is a coordinate plane ($E$ may be the whole $\mathbb R^n$). Let us consider two cases:

	\begin{enumerate}
		\item Assume that $\gamma \notin E^\bot$. We obtain from the induction hypothesis that the system (\ref{check_sys})($E$) (which is of $dim(E)$ variables) has no solutions. The whole system (\ref{check_sys}) has therefore no solutions.

		\item Assume that $\gamma \in E^\bot$. Suppose the asymptotic $v$ to be raising from asymptotic $v_E^{(i)}$.
		By the induction hypothesis, the subsystem (\ref{check_sys})($E$) is generic and has a finite number of solutions in the corresponding torus. Consider its solution and plug it in the subsystem (\ref{check_sys})($E^\bot$). The obtained system has no solutions due to lemma \ref{kush_check}. The intersection of a finite number of nonempty Zariski open sets is a nonempty Zariski open set, so the conditions of genericity are applied in general position.
	\end{enumerate}

	\subsection{Simplicity}

	Let us prove a lemma first.

	\begin{lemma}\label{as_simp}
		Suppose $\As \subset \mathbb Z_{\ge 0}^n$ to be a finite set such that $dim(Conv(\As)) = n$. Then a polynomial $p\in L_\As$ has only Morse singularities in the torus in case of general position of the polynomial.
	\end{lemma}

	\begin{proof}
		We should, first of all, prove that there are $n$ monomials $q_1,\dots,q_n$ of polynomial $p$ such that $d(q_1), \dots, d(q_n)$ form a basis at every point of the torus.

		\newcommand{\q}{\mathbf q}

		The dimension $dim(Conv(\As)) = n$, so there are $n$ points $\q_1, \dots, \q_n$ of the support $\As$ such that vectors from the origin to them are spanning $\mathbb R ^ n$. Let us prove that the monomials whose exponents are equal to $\q_1,\dots, \q_n$ are the desired ones.
		\renewcommand{\e}{\mathbf e}

		As is well known, there is a basis of the lattice $\e_1, \dots, \e_n$, such that the following system is applied
		\begin{equation}
		\begin{cases}
		\q_1 = m_1 \e_1 \\
		\q_2 = m_{2,1} \e_1 + m_2 \e_2\\
		\dots \\
		\q_n = m_{n,1} \e_1 + m_{n,2} \e_2 + \dots + m_n \e_n
		\end{cases},
		\end{equation}

		where $m_1 \ne 0, m_2 \ne 0, \dots , m_n \ne 0$.

		Denote by $e_1,\dots, e_n$ the monomials with exponents $\e_1, \dots,\e_n$. Let us change the variables $x_1,\dots x_n$ to $e_1,\dots e_n$. After the change of the variables the statement that $q_1,\dots q_n$ form a basis of $1$-forms at every point can be obviously proved by induction on $n$.

		Fix all the monomials of polynomial $p$ except for the $q_1,\dots q_n$ ones. Let us prove that the polynomial $p + c_1 q_1 + c_2 q_2 + \dots + c_n q_n$ has only Morse singularities in the torus in general position of the vector $(c_1,\dots, c_n)$.

		The differentials of the monomials is a basis at every point, so the smooth map $\psi: (\mathbb C\setminus\{0\})^n \to \mathbb C^n$, sending a point $x\in (\mathbb C\setminus\{0\})^n$ to the vector $(c_1(x),\dots,c_n(x))$ such that the polynomial $p + c_1(x) q_1 + c_2(x) q_2 + \dots + c_n(x) q_n$ has a singularity at the point $x$, is well-defined. Critical points of $\psi$ are those $x$ for which the polynomial $p + c_1(x) q_1 + c_2(x) q_2 + \dots + c_n(x) q_n$ has not Morse singularity at $x$. From the Sard's lemma we obtain that the measure of critical values is zero, so for almost all $(c_1,\dots, c_n)$ every critical point of $p + c_1 q_1 + c_2 q_2 + \dots + c_n q_n$ in the torus is Morse.
	\end{proof}

	Let us prove the simplicity of asymptotics by induction on $n$. Let $v$ be an asymptotic with positive multiplicity. Consider the system

	\begin{equation}\label{sim_sys}
	\begin{cases}
	\pi_\e^*(h_1^{v_\e}) (x) = 0\\
	\dots\\
	\pi_\e^*(h_n^{v_\e}) (x) = 0
	\end{cases}
	\tag{$**$}
	\end{equation}
	Due to theorem \ref{loc_kush}, it is enough to check that the system (\ref{sim_sys}) has no multiple solutions in general position.

	Note that the subsystem (\ref{sim_sys})($E(v)$) has, by induction hypothesis, only simple solutions in the torus and there is a finite number of them. After plugging any of them in the subsystem (\ref{sim_sys})($E^\bot(v)$) we obtain from lemma \ref{as_simp} that all solutions of the system (\ref{sim_sys})($E^\bot(v)$) are simple in general position. All the asymptotics of the system (\ref{gen}) are therefore simple.

	\section{Non-negative analogue of the Kouchnirenko formula}\label{formula_sec}

	In the case of $h = f + \e g$ the sum of asymptotics multiplicities is a non-negative formula for the Newton numbers difference $\nu(\Gp) - \nu(\Fp)$. In the section we simplify the sum of asymptotics multiplicities of the generalized problem \rom{1} on critical points, in particular to obtain a non-negative analogue of the Kouchnirenko formula (\ref{N-n KF}).\\

	\begin{definition}
		Consider a trace polyhedron $\Tp (v_E^{(i)}) \subset E^\bot \oplus \mathbb R^1$. Denote by $\Tpp(v_E^{(i)})$ the Newton polyhedron of $\pi_{\mathbb R^1}(\Tp(v_E^{(i)})) \subset E^\bot$. The union of projections of the faces $\pi_{\mathbb R^1}(\Tp^v(v_E^{(i)})), v \in \t(v_E^{(i)})$ splits the cone $E_{\ge0}^\bot$ into 3 areas (the upper area, the image of the projection itself and the lower area which contains the origin). The upper area is called the \emph{essential polyhedron} $\Tpe(v_E^{(i)})$ of polyhedron $\Tp(v_E^{(i)})$.
	\end{definition}\\

	Note that $\Tpe(v_E^{(i)})$ is usually not a convex polyhedron. But we still can define the Newton number of it.\\

	\begin{definition}
		Suppose that a polyhedron $\Tpe \in \mathbb R^n_{\ge 0}$ is cocompact (i.e its complement $\mathbb R^n_{\ge 0} \setminus \Tpe$ is compact).
		\emph{Newton number $\nu(\Tpe)$ of the polyhedron} $\Tpe$ is the number obtained by substituting $\Tpe$ in the formula (\ref{KF}) (see p.\pageref{th_kush}). It is well-defined since the complement of $\Tpe$ in the positive octant in compact. Assume that $\Tpe \subset \Tpe_1$, then denote by $\nu( \Tpe, \Tpe_1)$ the difference $\nu(\Tpe) - \nu(\Tpe_1)$.
	\end{definition}\\

	Consider a covector $v \in E^*_{\ge 0}$, where $E \subseteq \mathbb R^n$ is a coordinate plane. We can define the trace polyhedron $\Tp_E(v)$ for the problem {\bfseries \rom{1}} of $dim(E)$ variables with the support $\Hs_E$ (for example, $\Tp_{\mathbb R^n}(v) \equiv \Tp(v)$). The polyhedra $\Tpp_E(v)$ and $\Tpe_E(v)$ are defined similarly. Relatively open pairwise disjoint coordinate cones $E_{>0}$ form a partition of nonnegative cone $\mathbb R^n_{\ge 0} \setminus \{0\}$. Assume that $v_E^{(i)} \in E^*_{>0}$, then denote by  $m_{v_E^{(i)}}$ the multiplicity of the asymptotic $v_E^{(i)}$ in the system (\ref{gen})$_E$.

	We are going to group all asymptotics and sum up their multiplicities step by step. So let us introduce some nomenclature for the groups. Symbols $I, S, R$ and $D$ are for internal, semi-internal, raised and dropped asymptotics respectively. \label{notations}

	By writing some of these symbols without specifying a coordinate plane we mean the union over all $E\subset \mathbb R^n$ of sets of all asymptotics of prescribed classes in the problem {\bfseries \rom{1}} for the support $\Hs_E$. For example, $IS$ is the set of all internal and semi-internal asymptotics in all coordinate subproblems.

	If the coordinate plane $E$ is specified, then the set of all asymptotics of prescribed classes in the problem {\bfseries \rom{1}} for the support $\Hs_E$ is considered. For example, $ISRD(\mathbb R^n)$ is the set of all asymptotics of the generalized problem {\bfseries \rom{1}} on critical points of a function.

	We set $\nu(\Tpe (v)) = 1$ for all $v \in \mathbb R^n_{>0}$ (just to simplify some further formulas).

	\begin{theorem} \label{formula}
		The sum of multiplicities of asymptotics of critical points of polynomial $h$ tending to the origin as $\e \to 0$ is equal to the sum

		\begin{equation}
		\sum_{v \in IS} \nu(\Tpe(v))m_{v}
		\tag{\ref{N-n KF}}
		\end{equation}

	In the case of  $h = f + \e g$ it equals the difference of the Milnor numbers of generic singularities with embedded Newton polyhedra $\Fp$ and $\Gp$.
	\end{theorem}

	Let us prove a lemma first. The theorem is a simple corollary of the lemma.

	Suppose  $\Ps \subset \mathbb Z^n_{\ge 0} \oplus \mathbb Z$ to be a support and $\Pp$ to be the Newton polyhedron of $\Ps$. Assume that the polyhedron $\Pp$ contains a vertex $\mathbf u$ on the $\e$-axis (the last coordinate). Denote by $\mathbf T$ the set of all points of the support $\Ps$ which do not belong to the $\e$-axis. Let $t\in L_\mathbf T$ be in general position and $\Tp \in \mathbb R^n_{\ge 0}\oplus \mathbb R^1$ be the Newton polyhedron of $t$. Let $\Tpe \in \mathbb R^n_{\ge 0}$ be the essential polyhedron of the polyhedron $\Tp$ (with respect to the projection along $\e$-axis). Let $\Tpp \in \mathbb R^n_{\ge 0}$ be the Newton polyhedron of the projection $\pi_\e (\Tp)$.

	Note that even for dropped asymptotics the face $\Pp ^v$ is also well-defined (we just replace the infinite coordinates with some sufficiently large numbers). Now we are prepared to formulate the lemma.

	\begin{lemma}\label{form_lem}
		The sum of the multiplicities of critical points of the polynomial $t$ with asymptotics $v:\Pp ^v = \mathbf u$ is equal to $\nu (\Tpe, \Tpp)$.
	\end{lemma}

	\begin{ut}\label{lemtoth}
	Assume the lemma holds up to dimension $n-1$. Then the theorem \ref{formula} is also true up to dimension $n$.
	\end{ut}

	\begin{proof}[Proof of the remark]

	Let us introduce a definition first.\\

	\begin{definition}\label{generally raising}
		An asymptotic $v$ is said to be \emph{generally raising from} asymptotic $v_E^{(i)}\in E^*_{>0}$ if $\Hp^v = \Hp_E^{v_E^{(i)}}$, except for the dropped asymptotic $v_E^{(i)} + v_E^\infty$. The difference from \emph{raising from} $v_E^{(i)}$ is that some dropped asymptotics are also included.
	\end{definition}\\

	From lemma \ref{form_lem} we obtain that the sum of multiplicities of generally raising from $v_E^{(i)}$ asymptotics equals $\nu\left[\Tpe (v_E^{(i)}), \Tpp (v_E^{(i)})\right] m_{v_E^{(i)}}$. It is known that the multiplicity of dropped asymptotic $v_E^{(i)} + v_E^\infty$ equals $\nu(\Tpp(v_E^{(i)})) m_{v_E^{(i)}}$ (see theorem \ref{lie_th}). The total is $\nu(\Tpe(v_E^{(i)})) m_{v_E^{(i)}}$ and having summed this up over all asymptotics $v \in IS$ we obtain theorem \ref{formula}.
	\end{proof}

	\begin{proof}[Proof of lemma \ref{form_lem}]
	 Let us prove the lemma by induction on $n$. From the inclusion-exclusion principle we obtain the following remark

	 \begin{note}
	 	\begin{equation}\sum_{E\subset \mathbb R^n} \nu((\Tpe \cap E, \Tpp \cap E) = V_N(\Tpp \setminus \Tpe),
	 	\end{equation}
	 	where $E$ runs over all positive dimensional coordinate subspaces of $\mathbb R^n$.
	 \end{note}

 	Lemma \ref{form_lem} can be therefore restated. It is enough, by induction hypothesis, to prove that

 	\begin{equation}
 	\sum_{v \in ISRD} m_v = V_N(\Tpp \setminus \Tpe)
 	\end{equation}

 	The idea of the proof is as follows. We check that the difference between the volume $ V_N(\Tpp \setminus \Tpe)$ and the sum of the multiplicities of internal and semi-internal asymptotics $\sum\limits_{v\in IS(\mathbb R^n)}m_v$ equals the sum of the multiplicities in all the subproblem plus the sum of the multiplicities of dropped and raised asymptotics $\sum\limits_{v\in (ISRD\setminus ISRD(\mathbb R^n))}m_v + \sum\limits_{v\in RD(\mathbb R^n)}m_v$.

 	Note that the following equation holds
 	\begin{equation}
 	V_N(\Tpp \setminus \Tpe) - \sum_{v\in IS(\mathbb R^n)}m_v =  \sum_{v\in IS(\mathbb R^n)}(V_N(\Tp^v) - m_v) = \sum_{v\in S(\mathbb R^n)}( V_N(\Tp^v) - m_v)
 	\end{equation}

 	Consider an asymptotic $v \in S(\mathbb R^n)$. Denote by $\mathcal E (v)$ the set of all coordinate planes $E \subsetneq \mathbb R^n$ such that $dim(E\cap \Tp^v) = dim(E)$. Let $v_E^{(i)}$ be the orthogonal projection of $v$ onto $E$.
 	Due to formula (\ref{S-I}), the following equation is applied

 	\begin{equation}
 	m_v = V_N(\Tp^v) - \sum_{E\in \mathcal E(v)} V_N(\zp^{\Tp^v}_{E\cap \Tp^v})m_{v_E^{(i)}}
 	\end{equation}

 	So we can continue the previous equation

 	\begin{equation}\label{20}
 	\sum_{v\in S(\mathbb R^n)}( V_N(\Tp^v) - m_v) = \sum_{v \in S(\mathbb R^n)} \sum_{E \in \mathcal E(v)} V_N(\zp^{\Tp^v}_{E\cap \Tp^v})m_{v_E^{(i)}}
 	\end{equation}

 	Denote by $\Tp_{\downarrow}$ the complement $E_{\ge 0}\setminus \Tp$, where $\Tp \subset E$. Having grouped together all summands corresponding to $v_E^{(i)}$ we obtain an equivalent to (\ref{20}) expression

 	\begin{equation}\label{want_get}
 	\sum_{v\in(IS\setminus IS(\mathbb R^n))}   V_N(\Tpe_{\downarrow}(v)) m_v
 	\end{equation}

 	Our goal is to prove that it is equal to the sum

 	\begin{equation}
 	\sum_{v\in(RD(\mathbb R^n))} m_v + \sum_{v\in(ISRD\setminus ISRD(\mathbb R^n))} m_v
 	\end{equation}

 	Each asymptotic in the above sum is either in $IS \setminus IS(\mathbb R^n)$ or is generally raising from some asymptotic in $IS \setminus IS(\mathbb R^n)$. Thus we can rewrite the sum as

 	\begin{equation}
 	\sum_{v \in IS \setminus IS(\mathbb R^n)} (m_v +  \sum_{w\in GR(v)} m_w),
 	\end{equation}
 	where $GR(v)$ is the union of the sets of generally raising from $v$ asymptotics over all the corresponding subproblems. Due to Claim \ref{lemtoth}, this sum equals the following expression

 	\begin{equation}\label{24}
 	\sum_{E\subsetneq \mathbb R^n} \sum_{v_E^{(i)}\in IS(E)} m_{v_E^{(i)}}(1 + \sum_{E^\prime \supset E} \nu(\Tpe_{E^\prime}(v_E^{(i)})))
 	\end{equation}

 	From the inclusion-exclusion principle we obtain the following equality

 	\begin{equation}
 	1 + \sum_{E^\prime \supset E} \nu(\Tpe_{E^\prime}(v_E^{(i)})) = V_N(\Tpe_{\downarrow}(v))
 	\end{equation}

 	After substituting that in the previous expression (\ref{24}) we obtain the desired formula (\ref{want_get}).

	\end{proof}

	\begin{note}
		The proof of the lemma is combinatorial. For a linear morsification of a generic singularity the lemma states that the sum of the asymptotics multiplicities equals the Milnor number of a generic singularity calculated by the Kouchnirenko formula. We proved therefore combinatorially that the new formula is equivalent to the Kouchnirenko formula.
	\end{note}

	\section{Examples}\label{exs_sec}

	\subsection{Example for dimension 3 for problem \rom{1}}

	\graphicspath{{dim_3/}}

	Suppose support $\Fs$ to be as follows
	\begin{center}
		\begin{large}
			\begin{tabular}{ | c c   c  c  c c c|}
				\hline
				Point & $D$ & $E$ & $F$ & $G$ & $H$ & $I$ \\
				\hline
				Coordinate & $(0,5,0)$ & $(5,0,0)$ & $(4,1,0)$ & $(0,1,4)$ & $(0,0,6)$ & $(0,3,2)$ \\
				\hline
			\end{tabular}
		\end{large}\\
	\end{center}
	The support $\Gs$ (except for the points that are in $\Fs$) is as follows

	\begin{center}
		\begin{large}
			\begin{tabular}{ | c c   c  c |}
				\hline
				Point & $A$ & $B$ & $C$ \\
				\hline
				Coordinate & $(0,3,0)$ & $(2,2,0)$ & $(3,1,0)$ \\
				\hline
			\end{tabular}
		\end{large}\\
	\end{center}
\begin{figure}[H]
	\center{\includegraphics[scale=.5]{diags}}
	\caption{Newton polyhedra}
\end{figure}

	\begin{enumerate}
		\item \textbf{Internal asymptotics}

		There is only one internal asymptotic $\N(6,10,5) = (3,5,\frac{5}{2})$ belonging to a zero-dimensional polyhedral stratum. The dual of it is the tetrahedron $BEGH$. So its multiplicity equals $6V(BEGH) = 2$.

		\begin{figure}[H]
			\center{\includegraphics[scale=.5]{internel}}
			\caption{the tetrahedron $BEGH$ (brown)}
		\end{figure}

		\item \textbf{Semi-internal asymptotics}

		There are two semi-internal 3-dimensional polytopes $ABEG$ and $ADEG$.

		Consider the polytope $ABEG$ and the corresponding asymptotic $\N(8,12,7) = (2,3,\frac{7}{4})$.
		The local polytope of $z\frac{\partial (f + \e g)}{\partial z}$ at $(2,3,\frac{7}{4})$ is the point $G$. So the corresponding multiplicity equals $0$.

		\begin{figure}[H]
			\center{\includegraphics[scale=.5]{semi_2}}
			\caption{the tetrahedron $ABEG$ (yellow)}
		\end{figure}

		Consider the polytope $ADEG$ and the corresponding asymptotic $\N(1,1,1) = (\frac{1}{2},\frac{1}{2},\frac{1}{2})$. The polytope $ADEG$ contains 3 sutures $AD, ADE$ and $AGD$.

		\begin{figure}[H]
			\center{\includegraphics[scale=.5]{semi}}
			\caption{the tetrahedron $ADEG$ (yellow)}
		\end{figure}

		\begin{enumerate}
			\item Suture $AD$.

			$\tilde{\mathfrak{v}}_{AD} = l(AD) = 2$

			The polytope $\zp_{AD}^{ADEG}$ is triangle $p_{AD} (ICD) $, so $V_N(\zp_{AD}^{ADEG}) = 4$.

			\begin{figure}[H]
				\center{\includegraphics[scale=.5]{semi_AD}}
				\caption{another picture of the tetrahedron $ADEG$ (yellow)}
			\end{figure}

			\item Consider the suture $ADE$.

			\begin{equation}
			\tilde{\mathfrak{v}}_{ADE} = V_N(ADE) - V(AD,DF) = 2
			\end{equation}

			The segment $\zp_{ADE}^{ADEG}$ is $p_{ADE}(AI)$, the volume  $V_N(\zp_{ADE}^{ADEG})$ equals $2$.

			\item Consider the suture $ADG$.

			\begin{equation}
			\tilde{\mathfrak{v}}_{ADG} = V_N(ADG) - V(AD,DI) = 4
			\end{equation}

			The segment $\zp_{ADG}^{ADEG}$ is $p_{ADG}(AC)$, the volume  $V_N(\zp_{ADG}^{ADEG})$ equals $2$.
		\end{enumerate}

		Thus, the multiplicity of the asymptotic $(\frac{1}{2},\frac{1}{2},\frac{1}{2})$ equals the following expression

		\begin{equation}
		\begin{matrix}
		V_N(ADEG) - \tilde{\mathfrak{v}}_{AD} \cdot V_N(\zp_{AD}^{ADEG}) - \tilde{\mathfrak{v}}_{ADE} \cdot V_N(\zp_{ADE}^{ADEG}) -\\- \tilde{\mathfrak{v}}_{ADG} \cdot V_N(\zp_{ADG}^{ADEG}) = 40 - 2 \cdot 4 - 2\cdot 2 - 4\cdot 2 = 20
		\end{matrix}
		\end{equation}

		\item \textbf{Dropped asymptotics}

		The list of all possible dropped asymptotics:

		$$ (+\infty,\frac{1}{2},+\infty) \ \ \ (3,5,+\infty)\ \ \ (\frac{1}{2},\frac{1}{2},+\infty)\ \ \ (\frac{3}{4},\frac{1}{2},+\infty)\ \ \ (+\infty,\frac{1}{2},\frac{1}{2})$$

		Let us discuss all these cases:
		\begin{enumerate}
			\item Consider the asymptotic $(+\infty,\frac{1}{2},+\infty)$.

			\begin{figure}[H]
				\center{\includegraphics[scale=.5]{proj}}
				\caption{polytope $p_{Oy}(\Gs \cup \Fs)$}
			\end{figure}

		 	After removing the origin we obtain a support of a Morse function. The multiplicity is therefore $|AD|\cdot 1 = 2$.

			All the rest asymptotics have two finite coordinates. Suppose $v = v_E^{(i)} + v_E^\infty$ to be one of them. Then the multiplicity of asymptotic $v$ is equal to the multiplicity of $v_E^{(i)}$ in the subproblem associated with the plane $E$ multiplied by the decreased by one minimal value of the $E^\bot$ coordinate among all points of $\Fs \cup \Gs$ not belonging to $E$. For $xOy$ this point is $I$, its $z$-coordinate equals 2. For $yOz$ -- $C$, its $x$ coordinate is 2. Let us list all the other dropped asymptotics with their multiplicities.
			\item The asymptotic $(3,5,0)$ has multiplicity $1\cdot (2-1) = 1$.
			\item The asymptotic $(\frac{1}{2},\frac{1}{2},0)$ has multiplicity $2\cdot (2-1) = 2$.
			\item The asymptotic $(\frac{3}{4},\frac{1}{2},0)$ has multiplicity $4\cdot (2-1) = 4$.
			\item The asymptotic $(0,\frac{1}{2},\frac{1}{2})$ has multiplicity $4\cdot (2-1) = 4$.
		\end{enumerate}

		\item \textbf{Raised asymptotics}

		Asymptotics may be raising from an asymptotic of
		\begin{enumerate}
			\item the line $Oy$
			\item the plane $xOy$
			\item the plane $yOz$
		\end{enumerate}

		Let us discuss all these cases.

		\begin{enumerate}
			\item Consider asymptotics raising from the asymptotic $v_{Oy} = (0,\frac{1}{2},0)$.

			From the Figure \ref{1/2} below we obtain that the boundary of the essential polyhedron $ \Tpe(v_{Oy})$ is the projection of the segment $IF$ along $y$-axis onto the $xz$ plane, which is the segment $[(4,0,0), (0,0,2)]$, the Newton number therefore equals $3$. The points with accents are obtained from the initial ones by translation by $(0,2,0)$ in the picture below.

			\begin{figure}[H]
				\center{\includegraphics[scale=.5]{res_Oy}}
				\caption{\label{1/2}raising from $(0,\frac{1}{2},0)$ asymptotics}
			\end{figure}

			So the sum of multiplicities of asymptotics $v:\Hp^v = [AD]$ equals $|AD|\cdot 3 = 6$. We already know that the multiplicity of $(+\infty,\frac{1}{2},+\infty)$ equals $2$ and of $(\frac{3}{4},\frac{1}{2},+\infty)$ equals $4$. So, there are no more generally from $v_{Oy}$ asymptotics.

			\item Consider the plane $xOy$.

			\begin{figure}[H]
				\center{\includegraphics[scale=1.2]{xy}}
				\caption{the splitting of $\Gp_{xOy} \setminus \Fp_{xOy}$}
			\end{figure}

			Let us discuss all possible cases.

			\begin{enumerate}
				\item Consider asymptotics raising from the internal asymptotic $\N_{xOy}(3,5,0) = (3,5,0)$. The dual polytope is $ABE$.

				The only interesting face of the trace polyhedron corresponds to the segment $[IG]$. So the asymptotic is $(3,5,5)$, its multiplicity equals the doubled area of the triangle $ABE$ multiplied by the length of the projection of $[IG]$ onto $z$-axis, which is equal to $2\cdot \frac{1}{2}\cdot 2 = 2$.

				\item Consider asymptotics raising from the semi-internal asymptotic $\N_{xOy}(1,1,0) = (\frac{1}{2},\frac{1}{2},0)$. The dual polytope is $ADE$.

				The multiplicity of $(\frac{1}{2},\frac{1}{2},0)$ in $xOy$ equals 2.

				The difference $\Tpp (\frac{1}{2},\frac{1}{2},0) \setminus \Tpe (\frac{1}{2},\frac{1}{2},0)$ is empty. The multiplicity is therefore zero.

			\end{enumerate}

			\item Consider the plane $yOz$

			\begin{figure}[H]
				\center{\includegraphics[scale=1]{yz}}
				\caption{the splitting $\Gp_{yOz} \setminus \Fp_{yOz}$}
			\end{figure}

			Let us consider the only possible case.

			Consider asymptotics raising from $\N_{yOz}(0,1,1) = (0,\frac{1}{2},\frac{1}{2})$. The multiplicity of $(0,\frac{1}{2},\frac{1}{2})$ in $yOz$ equals 4.

			The only interesting face of the trace polyhedron corresponds to the segment $(C+(0,2,0), F)$.
			So the asymptotic $(\frac{3}{4},\frac{1}{2},\frac{1}{2})$ has multiplicity $4\cdot 2 = 8$.

		\end{enumerate}

	\end{enumerate}

		The final table of asymptotics and their multiplicities is as follows
		\begin{center}
		\begin{tabular}{ |p{35pt}|  c |  c | c |c|c|c|c|c|c|}
			\hline
			Asymp.
			& $(3,5, \frac{5}{2})$
			& $(\frac{1}{2},\frac{1}{2}, \frac{1}{2})$
			& $(3,5,5)$
			& $(\frac{3}{4},\frac{1}{2}, \frac{1}{2})$
			& $(0,\frac{1}{2},0)$
			& $(3,5,0)$
			& $(\frac{1}{2},\frac{1}{2},0)$
			& $(\frac{3}{4},\frac{1}{2},0)$
			& $(0,\frac{1}{2},\frac{1}{2})$ \\[7pt]
			\hline
			Mult.
			& $2$
			& $20$
			& $2$
			& $8$
			& $2$
			& $1$
			& $2$
			& $4$
			& $4$\\[7pt]
			\hline
		\end{tabular}
	\end{center}

	\subsection{Example for dimension 3 for the first non-degenerate jump of Milnor numbers}\label{jump_ex}

	\graphicspath{{dim3j/}}

	Consider Newton polyhedron $\Fp$ with the following set of vertices $\Fs$

	\begin{center}
			\begin{tabular}{ |c c   c  c c |}
				\hline
				Point & $A$ & $B$ & $C$ & $D$ \\
				\hline
				Coordinate & $(7,0,0)$ & $(0,7,0)$ & $(0,0,7)$ & $(2,2,2)$ \\
				\hline
			\end{tabular}
	\end{center}

	\begin{figure}[H]
		\center{\includegraphics[scale=.4]{polyh}}
		\caption{Newton polyhedron $\Fp$}
	\end{figure}

	From the Kouchnirenko formula we can obtain that $\mu_0 = \nu(\Fp) = 6\cdot6\cdot6 - 7\cdot7 = 167$. Let us prove that the first non-degenerate jump $\mu_0 - \mu_1$ (see p.\pageref{jumps}) is equal to $7$ in this case.
	We can consider one point expansions $\Gs = \{G\} \cup \Fs$ only to calculate the first jump. Let us consider three cases to prove that the first non-degenerate jump is $7$.

	\begin{enumerate}
		\item Assume that $G \in R^3_{>0}$. Then there is a bounded facet $K$ of $\Fs$ such that $\Gp \setminus \Fp$ contains the tetrahedron $Conv(K,G)$.

		\begin{figure}[H]
			\center{\includegraphics[scale=.4]{polyh3}}
			\caption{illustration for the case $G \in R^3_{>0}$}
		\end{figure}

		The difference of Newton numbers is therefore not less than $V_N(Conv(K,G)) \ge V_N(K) = 14 > 7$ as all the facets of the polyhedron $\Fp$ have normalized volume $14$.

		\item Assume that $G \in E_{>0}$, where $E$ is a 2-dimensional coordinate plane. Without loss of generality, assume $E$ to be the $xy$-plane. The normalized volume of the triangle $Conv(G,A,B)$ is at least $7$ and the essential polyhedron corresponding to this triangle is the ray $[2,+\infty)$. The difference of Newton numbers is therefore at least $7$. The difference $7$ is achieved if, for example, $G = (3,3,0)$.

		\begin{figure}[H]
			\center{\includegraphics[scale=.4]{polyh2}}
			\caption{illustration for the case $G \in xOy$}
		\end{figure}

		\item Assume that $G \in E_{>0}$, where $E$ is a 1-dimensional coordinate plane. Without loss of generality, assume $E$ to be the $x$-axis.

		\begin{figure}[H]
			\center{\includegraphics[scale=.5]{polyh1}}
			\caption{illustration for the case $G \in Ox$}
		\end{figure}

		The volume of the segment $[GA]$ is at least $1$. The corresponding essential polyhedron is as follows

		\begin{figure}[H]
			\includegraphics[scale=1]{ess}
			\caption{essential polyhedron}
		\end{figure}

		and its Newton Number is equal to $28-14+1 = 15$. The difference of the Newton numbers is hence at least $15 \cdot 1 = 15$ which is greater than $7$.
	\end{enumerate}

	The first non-degenerate jump is therefore equals $7$ in this example.

	\subsection{First example for dimension 4 for Arnold's problem on monotonicity}

	We can use the results of the paper to check whether two generic polynomials with the prescribed embedded Newton polyhedra have the same Newton number or not. If all the multiplicities of critical points of the corresponding line perturbation are zero, then the polyhedra define the same Newton number.

	Let us consider an example with $\Fs$ consisting of the following points
	\begin{center}
		\begin{small}
			\begin{tabular}{ |c c   c  c  c c c c c|}
				\hline
				Point & $A$ & $B$ & $C$ & $D$ & $E$ & $F$ & $G$ & $H$ \\
				\hline
				Coord. & $(21,0,0,0)$ & $(0, 21,0,0)$ & $(0,0,5,0)$ & $(0,0,0,5)$ & $(0,2,1,0)$ & $(2,0,0,1)$ & $(1,1,1,0)$ & $(1,1,0,1)$ \\
				\hline
			\end{tabular}
		\end{small}
\end{center}
	The support $\Gs$ contains one additional point $W = (10,10,0,0)$.

	The polytope $\Gp \setminus \Fp = Conv(A,B,E,F,G,H,W)$. There are no allowable (see definition on p.\pageref{allow}) zero-dimensional polyhedral strata. There is only one semi-internal zero-dimensional polyhedral stratum. It is the dual of $\Gp \setminus \Fp$.

	Let us prove that there are neither raised nor dropped asymptotics. The polytope $\Gp \setminus \Fp$ has appropriate intersections (which means that the dimension of the intersection equals the dimension of the plane) only with the planes $xyz$, $xyt$ and $xy$. And there is only one asymptotic in each corresponding subproblem.

	Pictures of the intersections of $\Gp \setminus \Fp$ with the hyperplanes $xyz$ and $xyt$ are below. The duals of semi-internal asymptotics are colored yellow

	\graphicspath{{dim_4.1/}}

	\begin{figure}[H]
		\center{\includegraphics[scale=.6]{xyz}}
		\caption{the intersections of $\Gp \setminus \Fp$ with $xyz$}
	\end{figure}

	\begin{figure}[H]
		\center{\includegraphics[scale=.6]{xyt}}
		\caption{the intersections of $\Gp \setminus \Fp$ with $xyt$}
	\end{figure}

	There are no raised asymptotics since all the differences $\Tpp \setminus \Tpe$ are empty.

	There are no dropped asymptotics since all the projections of $\Gs$ along the corresponding planes contain a point in the corresponding simplexes (see corollary on p. \pageref{no_lie}).

	Only the semi-internal asymptotic remains. Its multiplicity equals the mixed volume of the local polytopes, that is $MV(Conv(X,F,G,H), Conv(B,E,G,H), [EG],[FH])$. Note that the segments $[EG]$ and $[FH]$ are parallel, so the mixed volume is zero.

	Thus, adding the point $W$ to the support $\Fs$ does not affect the Newton number.

	\subsection{Second example for dimension 4 for Arnold's problem on monotonicity}
	The support $\Fs$ is as follows
	\begin{center}
		\begin{tabular}{ | c   c  c  c c|}
			\hline
			$A$ & $B$ & $C$ & $D$ & $E$ \\
			\hline
			$(10,0,0,0)$ & $(0,10,0,0)$ & $(0,0,10,0)$ & $(0,0,0,10)$ & $(5,1,0,0)$ \\
			\hline
		\end{tabular}\\
		\begin{tabular}{| c c c c|}
			\hline
			$F$ & $G$ & $H$ & $I$ \\
			\hline
			$(5,0,1,0)$ & $(5,0,0,1)$ & $(0,1,1,0)$ & $(0,0,1,1)$ \\
			\hline
		\end{tabular}\\
	\end{center}

	The support $\Gs$ contains the additional point $W = (9,0,0,0)$.\\
	\graphicspath{{dim_4.2/}}

	The intersection of $\Gp \setminus \Fp = Conv(A,E,F,G,H,I,W)$ with coordinate hyperplanes are as follows

	\begin{figure}[H]
		\center{\includegraphics[scale=.4]{xyz}}
		\caption{the intersections of $\Gp \setminus \Fp$ with $xyz$}
	\end{figure}

	\begin{figure}[H]
		\center{\includegraphics[scale=.4]{xyt}}
		\caption{the intersections of $\Gp \setminus \Fp$ with $xyt$}
	\end{figure}

	\begin{figure}[H]
		\center{\includegraphics[scale=.4]{xzt}}
		\caption{the intersections of $\Gp \setminus \Fp$ with $xzt$}
	\end{figure}

	There are only 3 asymptotics with nonzero multiplicities in the subproblems. There is one asymptotic in each of the hyperplanes $xyz$, $xzt$ and an asymptotic on the $x$-axis. But there are neither dropped nor raised asymptotics for the same reason as in the previous example.

	There are also no allowable internal zero-dimensional polyhedral strata.

	The only zero-dimensional semi-internal polyhedral stratum is the dual of $\Gp \setminus \Fp$.  Its multiplicity is the mixed volume of the local polytopes

	\begin{equation}
	MV(Conv(A,W,E,F,G),[EH], Conv(F,H,I), [GI])
	\end{equation} The segments $[EH]$ and $[GI]$ are parallel, so the mixed volume is zero.

	Thus, adding the point $W$ to the support $\Fs$ does not affect the Newton number.

config: arxiv