text:
\newcommand{\DBI}{\scrD^I_{\scrB}}
\newcommand{\DBprimeI}{\scrD^I_{\scrB'}}

\newcommand{\dUA}{\mscrD(U,\scrA)}
\newcommand{\dUUA}[1]{\mscrD(#1,\scrA)}
\newcommand{\dUAstar}{\mscrD^*(U,\scrA)}
\newcommand{\dUB}{\mscrD(U,\scrB)}
\newcommand{\dUBstar}{\mscrD^*(U,\scrB)}
\newcommand{\DUI}{\scrD^I_U}
\newcommand{\iUA}{\mscrI(U,\scrA)}
\newcommand{\iUUA}[1]{\mscrI(#1,\scrA)}
\newcommand{\iUAstar}{\mscrI^*(U,\scrA)}
\newcommand{\iUB}{\mscrI(U,\scrB)}
\newcommand{\iUBstar}{\mscrI^*(U,\scrB)}

\newcommand{\exc}{\mscrE}
\newcommand{\excbarA}{\bar \exc(\scrA)}
\newcommand{\excbarAA}[1]{\bar \exc(#1)}
\newcommand{\excA}{\exc(\scrA)}
\newcommand{\excU}{\exc(U)}
\newcommand{\excUA}{\exc(U,\scrA)}
\newcommand{\excUAI}{\exc^I(U,\scrA)}
\newcommand{\excUAprimeI}{\exc^I(U,\scrA')}

\newcommand{\multPstarS}{\multpstar{\scrP_1}{\scrP_n}{\mscrS}}

\newcommand{\VstarI}{(V^*)^I}
\newcommand{\VstarIf}{(V^*)^I(f)}

\section{Introduction}

In this section we compute the number of solutions on $\kk^n$ (or more generally, on any given Zariski open subset of $\kk^n$) of generic systems of polynomials with given supports, and give explicit BKK-type characterizations of genericness in terms of initial forms of the polynomials. As a special case we derive genralizations of weighted (multi-homogeneous)-B\'ezout theorems involving arbitrary weighted degrees (i.e.\ weighted degrees with possibly negative or zero weights).

\section{The bound}

\subsection{Khovanskii's formula} \label{first-section}
For polynomials $f_1, \ldots, f_n$, and any Zariski open subset $U$ of $\kk^n$, as in \cref{bkk-section} let $\multfiso{U}$ be the sum of intersection multiplicities of $f_1, \ldots, f_n$ at all the isolated points of $V(f_1, \ldots, f_n) \cap U$. Given a collection $\scrA := (\scrA_1, \ldots, \scrA_n)$ of $n$ finite subsets of $\znzero$, define

\begin{align*}
\multAiso{U} &:= \max\{\multfiso{U}:  \supp(f_j) \subset \scrA_j,\ j = 1, \ldots, n\}

\end{align*}
In this section we give a formula for $\multAiso{U}$ in terms of (mixed volumes of) convex hulls of $\scrA_j$. For $I \subseteq [n]$, let $\tiA := \{j: \Ai_j \neq \emptyset\}$ as in \cref{multiplicity-thm}, and let
\begin{align}
\excA := \{I \subseteq [n]:\ \text{there is $\tilde I \supseteq I$ such that $|\titildeA| < |\tilde I|$}\} \label{excA}
\end{align}

The following result, which follows immediately from \cref{thm:pure-dimension}, implies that $\multAiso{U} = \multAiso{\ua}$, where $\ua := U\setminus \bigcup_{i \in \excA} \Ki$.

\begin{lemma} \label{non-isolated-lemma}
Let $f_1, \ldots, f_m \in \kk[x_1, \ldots, x_n]$ and let $V := V(f_1, \ldots, f_m) \subset \kk^n$. Given $I \subseteq [n]$, if there exists $\tilde I \supseteq I$ such that $|\{j: f_j|_{\Kitilde} \not\equiv 0\}| < |\tilde I|$, then no point of $V \cap \Ki$ is isolated in $V$. \qed
\end{lemma}

Define
\begin{align}
\excU  &:= \{I \subseteq [n]: \Ki \cap U = \emptyset\}, \label{excU}\\
\touchUA & := \{I \subseteq [n]: I \notin \excU \cup \excA,\ |\tiA| = |I|\}. \label{touchUA}
\end{align}

The following result is also a straightforward consequence of \cref{thm:pure-dimension}.

\begin{lemma} \label{isolated-lemma}
If $I \not\in \touchUA$, then $\Kstari \cap V(f_1, \ldots, f_n) \cap U = \emptyset$ for generic $f_1, \ldots, f_n$ such that $\supp(f_j) \subseteq \scrA_j$, $j =1 , \ldots, n$. \qed
\end{lemma}

\begin{rem} \label{empty-touch}
It is possible that $\emptyset$ (i.e.\ the empty set) is in $\touchUA$; this is the case if and only if the origin is in $U$ and $0 < \multAzero < \infty$ (see \cref{ex-bkk-0}).
\end{rem}

\begin{thm}[Khovanskii\footnote{A.\ Khovanskii described this unpublished formula to the author during the Askoldfest, 2017}] \label{extended-bkk-bound-0}
Let $\scrP_j$ be the convex hull of $\scrA_j$, $j = 1, \ldots, n$. For $I \subseteq [n]$, let $\scrP^I_j := \scrP_j \cap \rr^I$, and let $\pi_I: \rr^n \to \ri$ be the natural projection (as in \eqref{pi_I}). Then
\begin{align}
\multAiso{U}
	= \sum_{I \in \touchUA}
							\mv(\scrP^I_{j_1}, \ldots, \scrP^I_{j_{|I|}})
							\times
							\multzero{\pi_{[n]\setminus I}(\scrP_{j'_1})}{\pi_{[n]\setminus I}(\scrP_{j'_{n-|I|}})}
	\label{extended-bkk-formula-0}
\end{align}
where for each $I \in \touchUA$, $j_1 , \ldots, j_{|I|}$ are the elements of $\tiA$, and $j'_1, \ldots, j'_{n-|I|}$ are the elements of $[n]\setminus \tiA$, and $\multzero{\cdot}{\cdot}$ is defined as in \eqref{multAzero}.
\end{thm}

The interpretation of the right hand side of \eqref{extended-bkk-formula-0} is straightforward - for each $I \in \touchUA$, the corresponding summand counts with multiplicity the number of solutions on $\Kstari \cap U$ of generic systems supported at $\scrA_1 ,\ldots, \scrA_n$.

In the next section we present another formula which sometimes is more efficient, since it involves summing over elements from a proper subset of $\touchUA$.

\begin{example} \label{ex-bkk-0}
Let $\scrA_j$ be the support of $f_j$ from \cref{ex-mv',ex0}, and $U$ be a nonempty Zariski open subset of $\kk^3$. Then $\excA = \emptyset$, and

\begin{align*}
\touchUA &=
	\begin{cases}
	\{\{1,2,3\},\{3\}, \emptyset\} &\text{if $\origin \in U$ (Case 1),}\\
	\{\{1,2,3\},\{3\}\} &\text{if $\origin \not\in U$, but $U$ contains a point on the $z$-axis (Case 2),}\\
	\{\{1,2,3\}\} &\text{otherwise (Case 3).}
	\end{cases}
\end{align*}
In Case 3, identity \eqref{extended-bkk-bound-0} and \cref{ex-mv'} imply that
\begin{align*}
\multpisonodots{\scrA_1,\scrA_2,\scrA_3}{U}  = \mv(\scrP_1, \scrP_2, \scrP_3) = 5
\end{align*}
Since the projections of $\scrP_2$ and $\scrP_3$ onto the $(x,y)$-plane have nontrivial linear part, and $\scrP^{\{3\}}_1$ has integer length $1$ (see \cref{fig-ex-bkk-0}), identity \eqref{extended-bkk-bound-0} implies that in Case 2,
\begin{align*}
\multpisonodots{\scrA_1,\scrA_2,\scrA_3}{U}
	&= 5+ \mv(\scrP^{\{3\}}_1) \times \multpnodots{\pi_{\{1,2\}}(\scrP_2),\pi_{\{1,2\}}(\scrP_3)}{\origin}
	= 5+ 1\cdot 1
	= 6
\end{align*}
 Finally, in Case 1, identity \eqref{extended-bkk-bound-0}  and the computation from \cref{ex0} imply that
\begin{align*}
\multpisonodots{\scrA_1,\scrA_2,\scrA_3}{U}
	&= 6 + \multpnodots{\scrP_1,\scrP_2,\scrP_3}{\origin}
	= 6 + 6
	= 12
\end{align*}

\begin{figure}[h]
\begin{center}

\begin{tikzpicture}[scale=0.6]
\pgfplotsset{every axis/.append style = {view={\viewx}{\viewy}, axis lines=middle, enlargelimits={upper}}}
\begin{scope}
\begin{axis}
	\addplot3 [fill=blue,opacity=0] coordinates{(3,0,0) (0,3,0) (0,1,0) (1,0,0) (3,0,0)};
	\addplot3 [blue, line width=4pt] coordinates{(0,0,1) (0,0,2)};
\end{axis}
\draw (1,4.5) node {$\scrP^{\{3\}}_1$};
\end{scope}

\begin{scope}[shift={(\shiftone,0)}]
\begin{axis}
	\addplot3 [black] coordinates{(0,0,1) (0,0,2)};
	\addplot3 [fill=blue,opacity=\opazero, thick] coordinates{(3,0,0) (0,3,0) (0,1,0) (1,0,0) (3,0,0)};
\end{axis}
\draw (4,0.5) node {$\pi_{\{1,2\}}(\scrP_2)$};
\end{scope}

\begin{scope}[shift={(2*\shiftone,0)}]
\begin{axis}
	\addplot3 [black] coordinates{(0,0,1) (0,0,2)};
	\addplot3 [fill=blue,opacity=\opazero, thick] coordinates{(2,0,0) (0,2,0) (0,1,0) (1,0,0) (2,0,0)};
\end{axis}
\draw (4,0.5) node {$\pi_{\{1,2\}}(\scrP_3)$};
\end{scope}
\end{tikzpicture}
\end{center}
\caption{Computing $\mv(\scrP^{\{3\}}_1) \times \multpnodots{\pi_{\{1,2\}}(\scrP_2),\pi_{\{1,2\}}(\scrP_3)}{\origin}$}
\label{fig-ex-bkk-0}
\end{figure}

\end{example}

\subsection{A formula in the same spirit as the formula for generic intersection multiplicity} \label{second-section}
If $\nu$ is a weighted order on $\kk[x_1, \ldots, x_n]$, we say that $\nu$ is {\em centered at infinity} if $\nu(x_i) < 0$ for some $i \in [n]$. Given $I \subseteq [n]$, we say that $\nu$ is {\em centered at $\Ki$} if $\nu(x_i) \geq 0$ for each $i \in I$ and $\nu(x_{i'}) > 0$ for each $i' \in [n]\setminus I$. Given a collection $\mscrS$ of subsets of $[n]$, we denote by $\Kns$ the complement in $\kk^n$ of the coordinate subspaces $\Ki$ for all $I \in \mscrS$, i.e.\
\begin{align}

\Kns &:= \kk^n \setminus \bigcup_{I \in \mscrS} \Ki \label{Kns}
\end{align}
We write $\Vs$ for the union, over all $I \in \scrS$, of the sets of weighted orders centered at $\Ki$, and $\Vinfty$ for the set of weighted orders centered at infinity; the collection of primitive elements in $\Vs$ and $\Vinfty$ are denoted respectively as $\Vprimes$ and $\Vprimeinfty$.

\begin{example} \label{Kns-remark}
Taking $\mscrS = \emptyset$ gives $\Kns =  \kk^n$ and $\Vs= \emptyset$. If we take $\mscrS=\{\emptyset\}$, then $\Kns= \kk^n \setminus \{0\}$ and $\Vs$ is the set $\Vzero$ of weighted orders {\em centered at the origin} (see \cref{mult-bound-statement-section}). If $\mscrS$ is the set of all subsets of $[n]$ consisting of $n-1$ elements, then $\Kns = \nktorus$ and $\Vs$ is the set of all nonzero weighted orders which are not centered at infinity.
\end{example}

Let $\scrA := (\scrA_1, \ldots, \scrA_n)$ be a collection of subsets of $\zzeroo{n}$ and $\scrP_j$ be the convex hull in $\rr^n$ of $\scrA_j$, $j = 1, \ldots, n$. Given a collection $\mscrS$ of subsets of $[n]$, define
\begin{align}
\multPstarS
	&:=  -\sum_{\nu \in \Vprimes \cup \Vprimeinfty} \min_{\scrP_1}(\nu) ~
		\mv'_\nu(\In_\nu(\scrP_2), \ldots, \In_\nu(\scrP_n))
	\label{multPstar}
\end{align}
where $\mv'_\nu(\cdot, \ldots, \cdot)$ is defined as in \eqref{mv'}.

\begin{thm}[{\cite{toricstein}}]\label{extended-bkk-bound}
Let $U$ be a Zariski open subset of $\kk^n$. We continue to use the notation of \cref{extended-bkk-bound-0}. Define $\touchUAone  := \{I \in \touchUA:  1 \in \tiA\}$, and for each $I \subseteq [n]$, set $\excUAI := \{J \subseteq I: J \in \excU \cup \excA\}$. Then
\begin{align}
\multAiso{U}
	= \sum_{I \in \touchUAone}
							\multpstar{\scrP^{I}_1, \scrP^{I}_{j_2}}{\scrP^{I}_{j_{|I|}}}{\excUAI}
							\times
							\multzero{\pi_{[n]\setminus I}(\scrP_{j'_1})}{\pi_{[n]\setminus I}(\scrP_{j'_{n-|I|}})}
	\label{extended-bkk-formula}
\end{align}
where for each $I \in \touchUAone$, $j_1 = 1, j_2, \ldots, j_{|I|}$ are the elements of $\tiA$, and $j'_1, \ldots, j'_{n-|I|}$ are the elements of $[n]\setminus \tiA$.
\end{thm}

There is an obvious analogy between formula \eqref{extended-bkk-formula} and the formula \eqref{mult-formula} for intersection multiplicity at the origin. The interpretations of the terms on the right hand side of \eqref{extended-bkk-formula} are also analogous to the interpretations of the corresponding terms of \eqref{mult-formula} described in \cref{mult-motivation}.  As in the case of \eqref{mult-formula}, and unlike \eqref{extended-bkk-formula-0}, the symmetry of the right hand side of \eqref{extended-bkk-formula} with respect to permutations of the $\scrP_j$ is not at all obvious. We use \eqref{extended-bkk-formula} to derive the symmetric formula of Huber and Sturmfels \cite{hurmfels-bern} and Rojas \cite{rojas-toric} in \cref{hsr-affine}.

\def\shiftx{6}

\def\colorzero{blue}
\def\opazero{0.5}
\def\viewx{70}
\def\viewy{5}
\def\titlex{1}
\def\titley{-1}
\def\scalefactor{0.6}
\def\colorzero{blue}
\def\colorone{green}

\begin{figure}[h]
\begin{center}
\begin{tikzpicture}[scale=0.6]
\pgfplotsset{every axis/.append style = {view={\viewx}{\viewy}, axis lines=middle, enlargelimits={upper}}}

\begin{scope}
\begin{axis}
	\addplot3 [fill=blue,opacity=\opazero, thick] coordinates{(3,0,0) (0,3,0) (0,1,4) (1,0,4) (3,0,0)};
	\addplot3 [fill=red,opacity=\opazero, thick] coordinates{(3,0,0) (1,0,4) (1,0,3) (2,0,1) (3,0,0)};
	\addplot3 [dashed, black, thick] coordinates{(1,0,3) (0,1,3) (0,1,4) (1,0,4) (1,0,3)};
	\addplot3 [dashed, black, thick] coordinates{(1,0,3) (0,1,3) (0,2,1) (2,0,1) (1,0,3)};
	\addplot3 [dashed, black, thick] coordinates{(0,2,1) (2,0,1) (3,0,0) (0,3,0) (0,2,1)};
\end{axis}
\draw (\titlex,\titley) node {$\scrP_1 + \scrP_3$};
\end{scope}

\def\factorone{4.5}
\def\factortwo{4.5}
\def\factorthree{1}
\begin{scope}[shift={(\shiftx,0)}]
\begin{axis}[hide axis, enlargelimits={0.15}]
	\addplot3 [ultra thick, ->] coordinates{(0,0,0) (2,2,1)};
	\addplot3 [ultra thick, blue, ->] coordinates{(0,0,0) (-2*\factorthree,-2*\factorthree,-\factorthree)};
	\addplot3 [ultra thick, ->] coordinates{(0,0,0) (1,1,1)};
	\addplot3 [ultra thick, ->] coordinates{(0,0,0) (\factortwo,\factortwo,0)};
	\addplot3 [ultra thick, red, ->] coordinates{(0,0,0) (0,\factorone,0)};
	\addplot3 [ultra thick, ->] coordinates{(0,0,0) (\factortwo,0,0)};
	\draw (axis cs:2,2,1) node [right] {(2,2,1)};
	\draw (axis cs:1,1,1) node [above] {(1,1,1)};
	\draw (axis cs:0,\factorone,0) node [right] {(0,1,0)};
	\draw (axis cs: \factortwo,0,0) node [below] {(1,0,0)};
	\draw (axis cs:\factortwo,\factortwo,0) node [below] {(1,1,0)};
	\draw (axis cs:-2*\factorthree,-2*\factorthree,-\factorthree) node [below] {(-2,-2,-1)};
\end{axis}

\draw (\titlex,\titley) node [right, text width=3.5cm] {inner normals to facets of $\scrP_1 + \scrP_3$};
\end{scope}

\end{tikzpicture}
\end{center}
\caption{$\scrP_1+ \scrP_3$ and the inner normals to its facets} \label{fig-ex-bkk-1-1}
\end{figure}

\begin{example}\label{ex-bkk-1}
We continue with $\scrA_1, \scrA_2, \scrA_3$ from \cref{ex-bkk-0}, and compute $\multpisonodots{\scrA_1,\scrA_2,\scrA_3}{U} $ using \cref{extended-bkk-bound} for nonempty Zariski open subsets $U$ of $\kk^3$. It is straightforward to check that
\begin{align*}
\touchUAone
	=
	\begin{cases}
	\{\{1,2,3\},\{3\}\} &\text{if $U$ contains a point on the $z$-axis (Case 1 or 2 of \cref{ex-bkk-0}),}\\
	\{\{1,2,3\}\} &\text{otherwise (Case 3 of \cref{ex-bkk-0}).}
	\end{cases}
\end{align*}
On the other hand, if we change the order of the $\scrA_j$, or equivalently, consider the collection $\scrA' := (\scrA_2, \scrA_1, \scrA_3)$, then one checks that $\touchUAprimeone =\{ \{1,2,3\}\}$ for any (nonempty) $U$, and we apply identity \eqref{extended-bkk-formula} to $\scrA'$ to reduce computation. In particular, we have
\begin{align*}
\multpisonodots{\scrA_1,\scrA_2,\scrA_3}{U}
	&= \multpisonodots{\scrA_2,\scrA_1\scrA_3}{U}
	= \multpstarnodots{\scrP_2, \scrP_1, \scrP_3}{\excUA}
	= -\sum_{\nu \in \Vprimess{\excUA} \cup \Vprimeinfty} \min_{\scrP_2}(\nu) ~
			\mv'_\nu(\In_\nu(\scrP_1), \In_\nu(\scrP_3))
\end{align*}

\begin{figure}[h]
\begin{center}
\begin{tikzpicture}[scale=0.4]

\def\shiftone{4}
\def\shifttwo{5}
\def\nux{-0.5}
\def\nuy{3.5}
\def\plusy{1.5}
\def\opazero{0.5}
\def\tx{-0.5}
\def\ty{-0.5}
\def\gridx{3.5}
\def\gridxx{3.5}
\def\gridy{2.5}
\def\bigshiftdhor{3}
\def\bigshiftdver{3}

\draw (\nux,\nuy) node [right] {\picfontsize $\nu = (-2,-2,-1)$};
\draw (\shiftone,\plusy) node {$+$};
\draw (\shiftone+\shifttwo,\plusy) node {$=$};

\draw [gray,  line width=0pt] (-0.5,-0.5) grid (\gridx,\gridy);

\draw[ultra thick, fill=\colorzero, opacity=\opazero ] (0,0) -- (1,0) -- (0,1) -- cycle;
\draw (\tx,\ty) node [below right] {\picfontsize $\In_\nu(\scrP_1)$};

\begin{scope}[shift={(\shifttwo,0)}]
	\draw [gray,  line width=0pt] (-0.5,-0.5) grid (\gridx,\gridy);

	\draw[ultra thick, fill=\colorzero, opacity=\opazero] (0,0) -- (2,0) -- (1,1) -- (0,1) --cycle;
	\draw (\tx,\ty) node [below right] {\picfontsize $\In_\nu(\scrP_3)$};
\end{scope}

\begin{scope}[shift={(2*\shifttwo,0)}]
	\draw [gray,  line width=0pt] (-0.5,-0.5) grid (\gridxx,\gridy);

	\draw[ultra thick] (0,0) -- (3,0) -- (1,2) --  (0,2) --  cycle;
	\draw[fill=\colorone, opacity=\opazero] (2,0) -- (1,1) -- (2,1) -- (3,0) -- cycle;
	\draw[fill=\colorone, opacity=\opazero] (0,1) -- (1,1) -- (1,2) -- (0,2) -- cycle;
	\draw[fill=\colorzero, opacity=\opazero, dashed, ultra thick] (0,0) -- (2,0) -- (1,1) -- (0,1) --cycle;
	\draw[fill=\colorzero, opacity=\opazero, dashed, ultra thick ] (1,1) -- (2,1) -- (1,2) -- cycle;

	\draw (\tx,\ty) node [below right] {\picfontsize $\In_\nu(\scrP_1 + \scrP_3)$};
\end{scope}

\begin{scope}[shift={(2*\shifttwo+\gridxx+\bigshiftdhor,0)}]
	\draw (\nux,\nuy) node [right] {\picfontsize $\nu = (2,2,1)$};
	\draw (\shiftone,\plusy) node {$+$};
	\draw (\shiftone+\shifttwo,\plusy) node {$=$};

	\draw [gray,  line width=0pt] (-0.5,-0.5) grid (\gridx,\gridy);

   \fill[blue] (0,0) circle (6pt);
	\draw (\tx,\ty) node [below right] {\picfontsize $\In_\nu(\scrP_1)$};

	\begin{scope}[shift={(\shifttwo,0)}]
		\draw [gray,  line width=0pt] (-0.5,-0.5) grid (\gridx,\gridy);

		\draw[ultra thick, fill=\colorzero, opacity=\opazero] (0,0) -- (2,0) -- (1,1) -- (0,1) --cycle;
		\draw (\tx,\ty) node [below right] {\picfontsize $\In_\nu(\scrP_3)$};
	\end{scope}

	\begin{scope}[shift={(2*\shifttwo,0)}]
		\draw [gray,  line width=0pt] (-0.5,-0.5) grid (\gridxx,\gridy);

		\draw[fill=\colorzero, opacity=\opazero, dashed, ultra thick] (0,0) -- (2,0) -- (1,1) -- (0,1) --cycle;

		\draw (\tx,\ty) node [below right] {\picfontsize $\In_\nu(\scrP_1 + \scrP_3)$};
	\end{scope}
\end{scope}

\begin{scope}[shift={(0,-\nuy-\bigshiftdver)}]
\draw (\nux,\nuy) node [right] {\picfontsize $\nu = (1,1,1)$};
\draw (\shiftone,\plusy) node {$+$};
\draw (\shiftone+\shifttwo,\plusy) node {$=$};

\draw [gray,  line width=0pt] (-0.5,-0.5) grid (\gridx,\gridy);

\draw[ultra thick, fill=\colorzero, opacity=\opazero ] (0,0) -- (1,0) -- (0,1) -- cycle;
\draw (\tx,\ty) node [below right] {\picfontsize $\In_\nu(\scrP_1)$};

\begin{scope}[shift={(\shifttwo,0)}]
	\draw [gray,  line width=0pt] (-0.5,-0.5) grid (\gridx,\gridy);

	\draw[ultra thick, blue] (0,0) -- (2,0);
	\draw (\tx,\ty) node [below right] {\picfontsize $\In_\nu(\scrP_3)$};
\end{scope}

\begin{scope}[shift={(2*\shifttwo,0)}]
	\draw [gray,  line width=0pt] (-0.5,-0.5) grid (\gridxx,\gridy);

	\draw[ultra thick] (0,0) -- (3,0) -- (2,1) --  (0,1) --  cycle;
	\draw[fill=\colorone, opacity=\opazero] (1,0) -- (3,0) -- (2,1) -- (0,1) -- cycle;
	\draw[fill=\colorzero, opacity=\opazero, dashed, ultra thick]  (0,0) -- (1,0) -- (0,1) -- cycle;

	\draw (\tx,\ty) node [below right] {\picfontsize $\In_\nu(\scrP_1 + \scrP_3)$};
\end{scope}

\begin{scope}[shift={(2*\shifttwo+\gridxx+\bigshiftdhor,0)}]
	\draw (\nux,\nuy) node [right] {\picfontsize $\nu = (1,1,0)$};
	\draw (\shiftone,\plusy) node {$+$};
	\draw (\shiftone+\shifttwo,\plusy) node {$=$};

	\draw [gray,  line width=0pt] (-0.5,-0.5) grid (\gridx,\gridy);

	\draw[ultra thick, blue] (0,0) -- (0,1);
	\draw (\tx,\ty) node [below right] {\picfontsize $\In_\nu(\scrP_1)$};

	\begin{scope}[shift={(\shifttwo,0)}]
		\draw [gray,  line width=0pt] (-0.5,-0.5) grid (\gridx,\gridy);

		\draw[ultra thick, blue] (0,0) -- (1,0);
		\draw (\tx,\ty) node [below right] {\picfontsize $\In_\nu(\scrP_3)$};
	\end{scope}

	\begin{scope}[shift={(2*\shifttwo,0)}]
		\draw [gray,  line width=0pt] (-0.5,-0.5) grid (\gridxx,\gridy);

		\draw[fill=\colorone, opacity=\opazero, ultra thick] (0,0) -- (1,0) -- (1,1) -- (0,1) --cycle;

		\draw (\tx,\ty) node [below right] {\picfontsize $\In_\nu(\scrP_1 + \scrP_3)$};
	\end{scope}
\end{scope}
\end{scope}
\end{tikzpicture}
\caption{The image under $\psi_\nu$ of initial faces of $\scrP_1, \scrP_3$ and $\scrP_1 + \scrP_3$}  \label{fig-ex-bkk-1-2}
\end{center}
\end{figure}

The inner normals to the facets of $\scrP_1 + \scrP_2$ are listed in \cref{fig-ex-bkk-1-1}. The only element in $\Vprimeinfty$ is $(-2,-2,-1)$. If the origin is in $U$, then $\excU = \emptyset$, and therefore it follows from \cref{mixed-example,fig-ex-bkk-1-2} that
\def\scalefactor{0.3}
\def\colorzero{blue}
\def\colorone{green}
\begin{align*}
\multpisonodots{\scrA_1,\scrA_2,\scrA_3}{U}
	&= - \min_{\scrP_2}(-2,-2,-1) \cdot
	\area(
		\begin{tikzpicture}[scale=\scalefactor]
			\draw[fill=\colorone, opacity=\opazero] (2,0) -- (1,1) -- (2,1) -- (3,0) -- cycle;
			\draw[fill=\colorone, opacity=\opazero] (0,1) -- (1,1) -- (1,2) -- (0,2) -- cycle;
		\end{tikzpicture}
		)
	= 6 \cdot 2
	=12
\end{align*}
If the origin is not in $U$, but $U$ contains some other points of the $z$-axis, then for $I = \{1,2,3\}$, the set $\excUAprimeI$ contains the emptyset, but not $\{3\}$. It follows that $\Vprimess{\excUAprimeI}$ does not contain $(1,1,0)$, but it contains each (primitive) weighted order centered at the origin (see \cref{Kns-remark}); in particular, it contains $(2,2,1)$ and $(1,1,1)$. Since $\min_{\scrP_2}(\nu) = 0$ when $\nu = (1,0,0)$ or $(0,1,0)$, it does not matter if these two elements are in $\Vprimess{\excUAprimeI}$. It then follows from \cref{fig-ex-bkk-1-2} that
\begin{align*}
\multpisonodots{\scrA_1,\scrA_2,\scrA_3}{U}
	&= 12
		-  \min_{\scrP_2}(2,2,1) \cdot
			\area(\emptyset	)
	 	- \min_{\scrP_2}(1,1,1) \cdot
			\area(
				\begin{tikzpicture}[scale=\scalefactor]
					\draw[fill=\colorone, opacity=\opazero] (1,0) -- (0,1) -- (2,1) -- (3,0) -- cycle;
				\end{tikzpicture}
				)
	= 12 - 4 \cdot 0 - 3 \cdot 2
	= 6
\end{align*}
If $U$ does not intersect the $z$-axis, then $(1,1,0)$ is also an element of $\Vprimess{\excUAprimeI}$, and it follows that
\begin{align*}
\multpisonodots{\scrA_1,\scrA_2,\scrA_3}{U}
	&= 6
	 	- \min_{\scrP_2}(1,1,0) \cdot
			\area(
				\begin{tikzpicture}[scale=\scalefactor]
					\draw[fill=\colorone, opacity=\opazero] (0,0) -- (1,0) -- (1,1) -- (0,1) -- cycle;
				\end{tikzpicture}
				)
	= 6  -  1 \cdot 1
	= 5
\end{align*}
The computations therefore agree with those from \cref{ex-bkk-0}. Note that formulae \eqref{extended-bkk-formula-0} and \eqref{extended-bkk-formula} resolve the cases in the opposite order!
\end{example}

\section{Derivation of the formuale for the bound} \label{extended-bkk-bound-section}
In this section we prove \cref{extended-bkk-bound-0,extended-bkk-bound}. Throughout this section $\scrA := (\scrA_1, \ldots, \scrA_n)$ denotes a collection of finite subsets of $\znzero$, and $\scrP_j$ denotes the convex hull of $\scrA_j$ in $\rr^n$, $j  = 1, \ldots, n$. As in the preceding chapters, we write $\scrL(\scrA)$ for the space of $n$-tuples of polynomials supported at $\scrA$, and as in \cref{strong-defn}, we write $\Nstrong(\scrA)$ be the collection of all strongly $\scrA$-non-degenerate $(f_1, \ldots, f_n) \in \scrL(\scrA)$. Given a Zariski open subset $U$ of $\kk^n$, let $\Nstrong(U,\scrA)$ for all $(f_1, \ldots, f_n) \in \Nstrong(\scrA)$ such that for all $I \not\in \excU \cup \excA$,
\begin{align}
(V(f_1, \ldots, f_n) \cap \Kstari) \setminus U = \emptyset \label{strong-U}
\end{align}
\Cref{A-non-degeneracy-existence,intersection-lemma} imply that $\Nstrong(U,\scrA)$ contains a nonempty Zariski open subset of $\scrL(\scrA)$. Assertion \eqref{openly-max} of \cref{mult-deformation-global} therefore implies that in order to prove \cref{extended-bkk-bound-0,extended-bkk-bound}, it suffices to show that $\multf{U}$ equals the quantities from the right hand sides of \eqref{extended-bkk-formula-0} and \eqref{extended-bkk-formula} for all $(f_1, \ldots, f_n) \in \Nstrong(U,\scrA)$.

\subsection{Proof of \cref{extended-bkk-bound-0}} \label{extended-bkk-bound-0-proof}

\Cref{extended-bkk-bound-0} follows from the following result.

\begin{prop}
Let $(f_1, \ldots, f_n) \in \Nstrong(U, \scrA)$ and $I \subseteq [n] \setminus (\excU \cup \excA)$.
\begin{enumerate}
\item \label{empty-I-assertion} If $I \not\in \touchUA$, then $V(f_1, \ldots, f_n) \cap \Kstari = \emptyset$.
\item \label{nonempty-I-assertion} If $I \in \touchUA$, then
\begin{align}
\sum_{a \in \Kstari} \multf{a}
	&= \mv(\scrP^I_{j_1}, \ldots, \scrP^I_{j_{|I|}})
							\times
							\multzero{\pi_{[n]\setminus I}(\scrP_{j'_1})}{\pi_{[n]\setminus I}(\scrP_{j'_{n-|I|}})}
	\label{bkk-formula-0-I}
\end{align}
where $j_1 , \ldots, j_{|I|}$ are elements of $\tiA$, and $j'_1, \ldots, j'_{n-|I|}$ are elements of $[n]\setminus \tiA$.
\end{enumerate}
\end{prop}

\begin{proof}
If $I \not\in \touchUA \cup \excA$, then $|\tiA| > |I|$, and property \ref{strongly-non-degenerate} of strongly $\scrA$-non-degenerate systems imply that $V(f_1, \ldots, f_n) \cap \Kstari = \emptyset$, which proves assertion \eqref{empty-I-assertion}. Now pick $I \in \touchUA$ and $a \in V(f_1, \ldots, f_n) \cap \Kstari$. Since proper non-degeneracy implies BKK non-degeneracy when dimension of the ambient affine space is equal to the number of polynomials, properties \ref{strongly-non-degenerate} and \ref{strong-np} of strongly $\scrA$-non-degenerate systems imply that $(f_j|_{\Ki}: j \in \tiA)$ are $(\Ai_j: j \in \tiA)$-non-degenerate, and \cref{finite-cor} implies that $ V(f_1, \ldots, f_n) \cap \Kstari$ is finite. If $j \in \tiA$, \cref{thm:pure-dimension} implies that $C := V(f_i: i\neq j) \cap \Kstari$ is purely one dimensional near $a$. Property \ref{intersectionally-non-degenerate} of strongly $\scrA$-non-degenerate systems and \cref{strong-cor} imply that $\Ki$ is an irreducible component of $V' := V(f_{j'}: j' \not\in \tiA)$, and no irreducible component of $V'$ other than $\Ki$ contains any irreducible component of $C$. \Cref{mult-chain-1} then implies that
\begin{align}
\multp{f_1}{f_n}{a} = \multp{f_{j_1}|_{\Ki}}{f_{j_{|I|}}|_{\Ki}}{a} \times \multzero{f_{j'_1, \epsilon}}
 	{f_{j'_{n-|I|}, \epsilon}} \label{bkk-formula-0-I-a}
\end{align}
for generic $\epsilon = (\epsilon_1, \ldots, \epsilon_n) \in \Kstari$, where $f_{j'_k, \epsilon}$ are formed from $f_{j'_k}$ by substituting $\epsilon_i$ for $x_i$ for all $i \in I$. Due to property \eqref{strong-U}, assertion \eqref{nonempty-I-assertion} follows by summing \eqref{bkk-formula-0-I-a} over all $a \in \Kstari$ due to \cref{bkk-non-degenerate-thm,non-degeneracy-0-thm} (after using $(\Ai_{j_1}, \ldots, \Ai_{j_{|I|}})$-non-degeneracy of $f_{j_1}|_{\Ki}, \ldots,  f_{j_{|I|}}|_{\Ki}$ and non-degeneracy at the origin of $f_{j'_1, \epsilon}, \ldots, f_{j'_{n-|I|}, \epsilon}$ for generic $\epsilon$).
\end{proof}

\subsection{Proof of \cref{extended-bkk-bound}}
Let $(f_1, \ldots, f_n) \in \Nstrong(U, \scrA)$ and $C$ be the subscheme defined by $f_2, \ldots, f_n$ on $U' := U \setminus \bigcup_{I\in \excA}\Ki$. For each $I \subseteq [n]$ such that $U' \cap \Ki \neq \emptyset$, let $\{C^I_j\}_j$ be the set of irreducible components of $C$ such that $\Ki$ is the smallest coordinate subspace of $\kk^n$ containing each $C^I_j$.

\begin{claim} \label{C^I_j}
Let $\touchUAone$ be as in \cref{extended-bkk-bound}, $I \subseteq [n]$ such that $U' \cap \Ki \neq \emptyset$, and $\ziA$ be as in \cref{strong-defn}.
\begin{enumerate}
\item If $\{C^I_j\}_j$ is nonempty, then $I \in \touchUAone$.
\item If $I \in \touchUAone$, then $[n] \setminus \tiA \in \ziA$.
\end{enumerate}
\end{claim}

\begin{proof}
For the first assertion, pick $I \subseteq [n]$ such that $\{C^I_j\}_j$ is nonempty. Since $I \not\in \excA$, it follows that $|\tiA| \geq |I|$. On the other hand, if $|\tiA \setminus \{1\}| \geq |I|$, then property \ref{strongly-non-degenerate} of strong $\scrA$-non-degeneracy and \cref{branch-lemma-IB} implies that each $\{C^I_J\}_j$ is a point, which contradicts \cref{thm:pure-dimension}. Accordingly $|\tiA \setminus \{1\}| = |I| - 1$ and $|\tiA| = |I|$, which imply that $I \in \touchUAone$, as required. The second assertion follows from the definition of $\excA$ and \cref{mult-support}.
\end{proof}

For each $I \in \touchUAone$, property \ref{strongly-non-degenerate} of strong $\scrA$-non-degeneracy and assertion \eqref{2-n-dim<n-1} of \cref{order-sum-nktorus} imply that either $\{C^I_j\}_j$ is empty, or each $C^I_j$ has dimension one. It then follows from \cref{C^I_j} that $C$ is a curve, and therefore \cref{cor:Macaulay,int-mult-curve,order-curve} imply that
\begin{align*}
\multf{U}
	&= \sum_{a \in C} \ord_a(f_1|_C) = \sum_{I,j,a} \multp{f_2}{f_n}{C^I_j}\ord_a(f_1|_{C^I_j})
\end{align*}
where $\multp{f_2}{f_n}{C^I_j}$ are defined as in \cref{complete-mult-section}. Pick $I \in \touchUAone$. Let $j_1 = 1, j_2, \ldots, j_{|I|}$ be the elements of $\tiA$, and $j'_1, \ldots, j'_{n- |I|}$ be the elements of $ [n] \setminus \tiA$. Property \ref{intersectionally-non-degenerate} of strongly $\scrA$-non-degenerate systems and \cref{strong-cor} imply that $\Ki$ is an irreducible component of $V' := V(f_{j'_1}, \ldots, f_{j'_{n-|I|}})$, and if $\{C^I_j\}_j$ is nonempty, then no irreducible component of $V'$ other than $\Ki$ contains any irreducible component of $C$. \Cref{mult-chain-0} then implies that
\begin{align*}
\multf{U}
	&= \sum_{I,j,a} \ord_a(f_1|_{C^I_j}) \multp{f_{j_2}|_{\Ki}}{f_{j_{|I|}}|_{\Ki}}{C^I_j} \times \multzero{f_{j'_1, \epsilon}}
		 	{f_{j'_{n-|I|}, \epsilon}}
\end{align*}
for generic $\epsilon = (\epsilon_1, \ldots, \epsilon_n) \in \Kstari$, where $f_{j'_k, \epsilon}$ are formed from $f_{j'_k}$ by substituting $\epsilon_i$ for $x_i$ for all $i \in I$. Properties \ref{recursively-non-degenerate}, \ref{strong-np} of strongly $\scrA$-non-degenerate systems and \cref{non-degeneracy-0-thm} imply that $\multzero{f_{j'_1, \epsilon}}{f_{j'_{n-|I|}, \epsilon}}
	= \multzero{\pi_{[n]\setminus I}(\scrP_{j'_1})}{\pi_{[n]\setminus I}(\scrP_{j'_{n-|I|}})}$ for generic $\epsilon \in \Ki$. In order to prove \cref{extended-bkk-bound} therefore it suffices to show that for each $I \in \touchUA$,
\begin{align*}
\sum_{j,a} \ord_a(f_1|_{C^I_j}) \multp{f_{j_2}|_{\Ki}}{f_{j_{|I|}}|_{\Ki}}{C^I_j}
	= 	\multpstar{\scrP^{I}_1, \scrP^{I}_{j_2}}{\scrP^{I}_{j_{|I|}}}{\excUAI}
\end{align*}
Identify $\Ki$ with $\kk^k$, where $k := |I|$. \Cref{order-nu-prop,C-Q} imply that we can find a $k$-dimensional convex rational polytope $\scrP$ such that
\begin{itemize}
\item $\Ki \into \xp$,
\item $V(f_{j_2}|_{\Ki}, \ldots, f_{j_k}|_{\Ki})$ extends to a complete curve $\bar C^I$ on $\xp$, and
\item $f_1$ restricts to a nonzero rational function on $\bar C^I$ which is representable near every point of $\bar C^I$ as a quotient of non zero-divisors.
\end{itemize}
\Cref{zero-sum} implies that
\begin{align*}
\sum_j \sum_{a \in U'} \ord_a(f_1|_{C^I_j}) \multp{f_{j_2}|_{\Ki}}{f_{j_k}|_{\Ki}}{C^I_j}
	= 	- \sum_j \sum_{a \in \bar C^I_j \setminus U'}  \ord_a(f_1|_{C^I_j}) \multp{f_{j_2}|_{\Ki}}{f_{j_k}|_{\Ki}}{C^I_j}
\end{align*}
where $\bar C^I_j$ are the closures of $C^I_j$ in $\xp$. Property \eqref{strong-U} of the $f_j$ implies that $a \in \bar C^I_j \setminus U'$ if and only if either $a \in \Kii{I'}$ for some $I' \in \excU \cup \excA$, or (the germ at) $a$ is a branch at infinity of $C^I_j$. \Cref{order-nu-cor} then implies that
\begin{align*}
\sum_j \sum_{a \in U'} \ord_a(f_1|_{C^I_j}) \multp{f_{j_2}|_{\Ki}}{f_{j_k}|_{\Ki}}{C^I_j}
	= 	- \sum_{\nu \in \Viprimess{\excUAI} \cup \Viprimeinfty} \min_{\scrP^I_1}(\nu) ~
			\mv'_\nu(\In_\nu(\scrP^I_{j_2}), \ldots, \In_\nu(\scrP^I_{j_k}))
\end{align*}
where $\Viprimeinfty$ (respectively $\Viprimess{\excUAI}$) is the set of primitive weighted orders on $\kk[x_i: i \in I]$ which are centered at infinity (respectively at $\Kii{I'}$ for some $I' \in \excUAI$). Since the right hand side of the preceding identity is precisely $\multpstar{\scrP^{I}_1, \scrP^{I}_{j_2}}{\scrP^{I}_{j_{|I|}}}{\excUAI}$, this completes the proof of \cref{extended-bkk-bound}.

\section{Other formulae for the bound} \label{other-section}
Throughout this section we continue to use $\scrA $ to denote a collection $(\scrA_1, \ldots, \scrA_n)$ of finite subsets of $\znzero$ and $\scrP_j$ to denote the convex hull of $\scrA_j$, $j = 1, \ldots, n$.

\subsection{The formula of Huber-Sturmfels and Rojas} \label{hsr-affine}
Let $t$ be a new indeterminate. Fix positive integers $k_1, \ldots, k_n$. Note that for each $f_1, \ldots, f_n \in \kk[[x_1, \ldots, x_n]]$, and each Zariski open subset $U$ of $\kk^n$,
\begin{align}
\multfiso{U}
	&= \multpisonodots{t, f_1 + c_1t^{k_1}, \ldots, f_n + c_nt^{k_n}}{U \times \kk} \label{lifted-system}
\end{align}
for any $c_1, \ldots, c_n \in \kk$. It follows that
\begin{align*}
\multAiso{U}
	&= \multpiso{\hat \scrA_0}{\hat \scrA_n}{\hat U}
\end{align*}
where $\hat U := U \times \kk$, $\hat \scrA_0 := \{(1, 0, \ldots, 0)\} \subset \zzeroo{n+1}$ and $\hat \scrA_j :=  \{(k_j, 0, \ldots, 0)\} \cup (\{0\} \times \scrA_j) \subset \zzeroo{n+1}$  for $j = 1, \ldots, n$. Let $\hat \scrA := (\hat \scrA_0, \ldots, \hat \scrA_n)$. It is straightforward to check that $\touchUUAAone{\hat U}{\hat \scrA} = \{[n+1]\}$, so that \cref{extended-bkk-bound} implies that
\begin{align*}
\multAiso{U}
	= \multpstar{\hat \scrP_0}{\hat\scrP_n}{\exc( \hat U)\cup \exc(\hat \scrA)}
	= - \sum_{\hat \nu \in \Vhatprimess{\exc(\hat U) \cup \exc(\hat \scrA)} \cup \Vhatprimeinfty}	\hat \nu_0 \mv'_{\hat \nu}(\In_{\hat \nu}(\hat \scrP_1), \ldots, \In_{\hat \nu} (\hat \scrP_n))

\end{align*}
where $\hat \scrP_j$ are the convex hulls of $\hat \scrA_j$, and $\hat \nu$ ranges over the collection $\Vhatprimess{\exc(\hat U) \cup \exc(\hat \scrA)} \cup \Vhatprimeinfty$ of all primitive weighted orders on $\kk[t, x_1, \ldots, x_n]$ which are either centered at infinity or centered at $\Ki$ for some $I \in \exc( \hat U)\cup \exc(\hat \scrA)$, and $\hat \nu_0 := \hat \nu(t)$. Now, since $\dim(\hat P_0) = 0$, either \cref{bkk-bound-thm} or \cref{positively-mixed} implies that $\mv(\hat \scrP_0, \ldots, \hat \scrP_n) = 0$. Therefore assertion \eqref{rational-mv'} of \cref{rational-mv'-prop} implies that
\begin{align}
\multAiso{U}
&= \sum_{\hat \nu \not\in \Vhatprimess{\exc(\hat U) \cup \exc(\hat \scrA)} \cup \Vhatprimeinfty}
	\hat \nu_0 \mv'_{\hat \nu}(\In_{\hat \nu}(\hat \scrP_1), \ldots, \In_{\hat \nu} (\hat \scrP_n))
\label{extended-bkk-formula-rojas-0}
\end{align}
If $\nu$ is a primitive weighted order on $\kk[t, x_1, \ldots, x_n]$, then $\nu \not\in \Vhatprimess{\exc(\hat U) \cup \exc(\hat \scrA)} \cup \Vhatprimeinfty$ if and only if all the following hold:
\begin{prooflist}
\item \label{non-neg} $\nu$ is nonnegative, and
\item \label{exc'} for each $I \in \excU \cup \excA$, there is $i' \not\in I$ such that $\nu(x_i) = 0$.
\end{prooflist}
Let $\Vhatprime(U,\scrA)$ be the set of all primitive weighted orders $\hat \nu$ on $\kk[t, x_1, \ldots, x_n]$ which satisfy properties \ref{non-neg}, \ref{exc'} and in addition satisfy the following:
\begin{prooflist}[resume]
\item  $\hat \nu_0 := \hat \nu(t)$ is positive.
\end{prooflist}
Since a summand on the right hand side of \eqref{extended-bkk-formula-rojas-0} has a nonzero contribution only if $\hat \nu_0$ is positive, it follows that for any Zariski open subset $U$ of $\kk^n$,
\begin{align}
\multAiso{U}
&= \sum_{\hat \nu \in \Vhatprime(U,\scrA)}
	\hat \nu_0 \mv'_{\hat \nu}(\In_{\hat \nu}(\hat \scrP_1), \ldots, \In_{\hat \nu} (\hat \scrP_n))
\label{extended-bkk-formula-rojas}
\end{align}
In the case that $\kk = \cc$, $\excA = \emptyset$, and $U = \kk^n \setminus V( \prod_{j \in J} x_j) \cong \nktoruss{|J|} \times \kk^{n-|J|}$ for some $J \subseteq [n]$, formula \eqref{extended-bkk-formula-rojas} appeared in \cite{hurmfels-bern}. In this case $\Vhatprime(U, \scrA)$ consists of all primitive nonnegative weighted orders $\hat \nu$ on $\kk[t, x_1, \ldots, x_n]$ such that $\hat \nu_0$ is positive, and $\hat \nu(x_j) = 0$ for each $j \in J$. The sum on the right hand side of \eqref{extended-bkk-formula-rojas} in this case was termed in \cite{hurmfels-bern} as the {\em $I$-stable mixed volume} (where $I := [n]\setminus J$) of $\scrA_1, \ldots, \scrA_n$. J.\ M.\ Rojas \cite{rojas-toric} observed that the formula of \cite{hurmfels-bern} works over all algebraically closed fields.

\subsection{Estimates in terms of single mixed volumes} \label{li-wang-section}
If $U$ is nonempty, identity \eqref{extended-bkk-formula-0} implies that $\multAiso{U} \geq \mv(\scrP_1, \ldots, \scrP_n)$. On the other hand, since $\scrA_j \subseteq \tilde \scrA_j := \scrA_j \cup \{\origin\}$, it trivially follows that $\multAiso{U} \leq \multAprimeiso{U} = \mv(\scrP'_1, \ldots, \scrP'_n)$, where $\scrP'_j$ are the convex hull of $\scrA'_j$, and the last equality follows from \eqref{extended-bkk-formula-0}. It follows that for nonempty Zariski open subsets $U$ of $\kk^n$,
\begin{align}
\mv(\conv(\scrA_1), \ldots, \conv(\scrA_n))
	\leq \multAiso{U}
	\leq \mv(\conv(\scrA_1 \cup \{\origin\}) \ldots, \conv(\scrA_n \cup \{\origin\}))
	\label{mv-estimate}
\end{align}
The upper bound in \eqref{mv-estimate} is due to T.\ Y.\ Li and X.\ Wang \cite{li-wang-bkk}. We now examine when these bounds are exact. The lower bound is easier to handle; the following result follows directly from \cref{positively-mixed,extended-bkk-bound-0}.

\begin{prop} \label{lower-prop}
Let $U$ be a nonempty Zariski open subset of $\kk^n$. Then the following are equivalent:
\begin{enumerate}
\item  The first inequality in \eqref{mv-estimate} holds with equality.
\item \label{lower-condition} For each $I \in \touchUA\setminus \{[n]\}$, $\scrP_{j_1} \cap \ri, \ldots, \scrP_{j_{|I|}} \cap \ri$ are {\em dependent}, where $j_1, \ldots, j_{|I|}$ are the elements of $\tiA$. \qed
\end{enumerate}
\end{prop}

\begin{rem}
Since an empty collection of convex polytopes is by definition {\em independent}, condition \eqref{lower-condition} of \cref{lower-prop} implies in particular that $\emptyset \not\in \touchUA$, which in turn implies that either $U$ does not contain the origin, or $\multAzero$ is zero or $\infty$ (\cref{empty-touch}).
\end{rem}

Following A.\ Khovanskii \cite{khovanus}, We say that $\scrA = (\scrA_1, \ldots, \scrA_n)$ is {\em regularly attached to the coordinate cross} if for each proper subset $I$ of $[n]$, the set of nonempty elements of $\{\scrP_j \cap \ri: j = 1, \ldots, n\}$ is dependent; in particular this implies (taking $I = \emptyset$) that the origin belongs to at least one of the $\scrA_j$. The following is immediate from \cref{lower-prop}.

\begin{cor}[Khovanskii {\cite{khovanus}}]
If $U$ is nonempty and $\scrA$ is regularly attached to the coordinate cross, then $\multAiso{U} = \mv(\scrP_1, \ldots, \scrP_n)$. \qed
\end{cor}

Now let $M := \mv(\conv(\scrA_1 \cup \{\origin\}) \ldots, \conv(\scrA_n \cup \{\origin\}))$ be the upper bound from \eqref{mv-estimate}. Consider as in \eqref{lifted-system} the system
\begin{align}
f_j  + t = 0,\ j = 1, \ldots, n, \label{lifted-system'}
\end{align}
where $f_j$ are generic polynomials supported at $\scrA_j$, with $\np(f_j) = \conv(\scrA_j) = \scrP_j$. For generic $t \neq 0$, the corresponding system has precisely $M$ isolated solutions, all of which are on $\nktorus$. Therefore, the number of solutions of the system at $t = 0$ is also $M$ if and only if there is no curve of solutions of the system \eqref{lifted-system'} on $\nktorus$ that escapes $U$ or becomes non-isolated as $t$ approaches $0$. \Cref{bkk-bound-thm,order-nu-prop,positively-mixed} imply that such a curve exists if and only if there is a weighted order $\hat \nu$ on $\kk[x_1, \ldots, x_n, t]$ such that
\begin{prooflist}
\item $\hat \nu(t) > 0$,
\item the restriction of $\hat \nu$ to $\kk[x_1, \ldots, x_n]$ is either centered at infinity or at $\Ki$ for some $I \in \excU \cup \excA$, and
\item $\In_{\hat \nu}(\hat \scrP_1), \ldots,  \In_{\hat \nu}(\hat \scrP_n)$ are {\em independent}, where $\hat \scrP_j := \np(f_j + t) \subset \rzeroo{n+1}$.
\end{prooflist}
Let $\nu$ be the restriction of $\hat \nu$ to $\kk[x_1, \ldots, x_n]$. Let $m_\nu := \max\{\min_{\scrP_j}(\nu): j = 1, \ldots, n\}$. Since $\dim( \In_{\hat \nu}(\sum_j \hat \scrP_j)) \leq \dim( \In_{\nu}(\sum_j \scrP_j)) + 1$, and since $\hat \nu(t) > 0$, it follows from the definition of dependence of polytopes that $\In_{\hat \nu}(\hat \scrP_1), \ldots,  \In_{\hat \nu}(\hat \scrP_n)$ are independent if and only if
\begin{prooflist}[resume]
\item  \label{m_nu-condition}$m_\nu > 0$, and
\item \label{dimension-condition} for all $J \subseteq [n]$,
\begin{align}
\dim(\In_\nu(\sum_{j \in J} \scrP_j))
	\geq
	\begin{cases}
	|J| &\text{if $\min_{\scrP_j}(\nu) < m_\nu$ for each $j \in J$,}\\
	|J| - 1 & \text{if there is $j \in J$ such that $\min_{\scrP_j}(\nu) = m_\nu$.}
	\end{cases}
	\label{li-wang-dimension-constraint}
\end{align}
\end{prooflist}
Taking $J = [n]$ in \eqref{li-wang-dimension-constraint} implies in particular that $\dim(\In_\nu(\sum_j \scrP_j)) = n-1$. These observations imply the following result.

\begin{prop} \label{li-wang-condition}
Let $\scrP := \sum_j \scrP_j$. The second inequality in \eqref{mv-estimate} holds with equality if and only if
\begin{enumerate}
\item \label{leq-n-2} either $\dim(\scrP) \leq n -2$, or
\item $\dim(\scrP) \geq n -1$, and for each face of dimension $(n-1)$ of $\scrP$ such that its primitive inner normal\footnote{If $\dim(\scrP) = n-1$, then both of the primitive normals to $\scrP$ are considered to be inner.} $\nu$ is centered at infinity or at $\Ki$ for some $I \in \excU \cup \excA$, at least one of the conditions \ref{m_nu-condition} and \ref{dimension-condition} fails for $\nu$. \qed
\end{enumerate}
\end{prop}

As a corollary we get a situation where {\em both} of the bounds of \eqref{mv-estimate} are exact.

\begin{cor} \label{conveniently-li-wang}
Assume each $\scrA_j$ is {\em convenient} and $U$ is a nonempty Zariski open subset of $\kk^n$. Assume in addition that at least one of the following holds:
\begin{enumerate}
\item either $U$ does not contain the origin,
\item or at least one of the $\scrA_j$ contains the origin.
\end{enumerate}
Then
\begin{align*}
 \multAiso{U}
 	= \mv(\conv(\scrA_1), \ldots, \conv(\scrA_n))
	=  \mv(\conv(\scrA_1 \cup \{\origin\}) \ldots, \conv(\scrA_n \cup \{\origin\}))
\end{align*}
\end{cor}

\begin{proof}
$\scrA_j$ satisfy the hypotheses of both \cref{lower-prop,li-wang-condition}.
\end{proof}

\section{Examples motivating the non-degeneracy conditions} \label{non-deg-example-section}

In this section we give some examples to motivate the necessary and sufficient conditions for the equality $\multfkniso = \multAkniso$, where $\scrA := (\scrA_1, \ldots, \scrA_n)$ is a given collection of finite subsets of $\znzero$, and $(f_1, \ldots, f_n) \in \scrL(\scrA)$. We consider the case $U = \kk^n$; in a sense this is the most important case, and it already captures the essence of the general case. We also assume for simplicity that $\excA = \emptyset$, which ensures in particular that for generic $(f_1, \ldots, f_n) \in \scrL(\scrA)$, all points in the set $V(f_1, \ldots, f_n)$ of common zeroes of $f_1, \ldots, f_n$ in $\kk^n$ are isolated. In this scenario, if we apply the intuitive reasoning from \cref{sec:bkk-non-degeneracy-statement} that motivated the non-degeneracy condition for Bernstein's theorem, we are led to the following condition:
\begin{align}
\parbox{0.69\textwidth}{$f_1|_{\Ki}, \ldots, f_n|_{\Ki}$ are {\em $\Ai$-non-degenerate at infinity} for each $I \subseteq [n]$.}
\tag{ND$_\infty$}
\label{non-deg-infinity}
\end{align}
where $\Ai := (\Ai_1, \ldots, \Ai_n) = (\scrA_1 \cap \ri, \ldots, \scrA_n \cap \ri)$, and {\em non-degeneracy at infinity} is defined as follows: given a collection $\scrB := (\scrB_1, \ldots, \scrB_m)$ of finite subsets of $\znzero$ and $g_j \in \kk[x_1, \ldots, x_n]$, such that $\supp(g_j) \subseteq \scrB_j$, $j = 1, \ldots, m$, we say that $g_1, \ldots, g_m$ are $\scrB$- non-degenerate at infinity if for each weighted order $\nu$ {\em centered at infinity} (see \cref{second-section}), there is no common root of $\In_{\scrB_j,\nu}(g_j)$, $j = 1, \ldots, m$, on $\nktorus$. We now present a series of examples which illustrate how condition \eqref{non-deg-infinity} falls short of characterizing the correct non-degeneracy condition, and which also suggest the ways to amend it. In all these examples $\scrP_j$ would denote the convex hull of $\scrA_j$, $j = 1, \ldots, n$.

\def\shiftx{9}

\def\colorzero{blue}
\def\opazero{0.5}
\def\viewx{70}
\def\viewy{5}
\def\titlex{3}
\def\titley{-1}
\def\scalefactor{0.6}
\def\colorzero{blue}
\def\colorone{green}

\begin{figure}[h]
\begin{center}
\begin{tikzpicture}[scale=0.6]
\pgfplotsset{every axis/.append style = {view={\viewx}{\viewy}, axis lines=middle, enlargelimits={upper}, axis equal}}

\begin{scope}
\begin{axis}
	\addplot3 [fill=blue,opacity=\opazero, thick] coordinates{(0,1,1) (1,1,0) (1,0,1)};
	\addplot3 [fill=red,opacity=\opazero, thick] coordinates{(0,0,0) (1,1,0) (1,0,1)};
	\addplot3 [thick, dashed] coordinates{(0,0,0) (0,1,1)};

\end{axis}
\draw (\titlex,\titley) node {$\scrP_1 = \scrP_2$};
\end{scope}

\begin{scope}[shift={(\shiftx,0)}]
\begin{axis}
	\addplot3 [fill=blue,opacity=\opazero, thick] coordinates{(0,1,2) (1,1,1) (1,0,2)};
	\addplot3 [fill=red,opacity=\opazero, thick] coordinates{(0,0,1) (1,1,1) (1,0,2)};
	\addplot3 [thick, dashed] coordinates{(0,0,1) (0,1,2)};

\end{axis}
\draw (\titlex,\titley) node {$\scrP_3 = \scrP_1 + \{(0,0,1)\}$};
\end{scope}

\end{tikzpicture}
\end{center}
\caption{Newton polytopes of \cref{ex-aff-non-deg-1}} \label{fig-ex-aff-non-deg-1}
\end{figure}

\begin{example}\label{ex-aff-non-deg-1}
Let $\scrA_1 = \scrA_2 = \{(0,0,0), (1,1,0), (0, 1, 1), (1,0,1)\} \subset \zzeroo{3}$ and $\scrA_3 = \scrA_1 + \{(0,0,1)\}$.  Then $\scrP_1 = \scrP_2$ is a tetrahedron and $\scrP_3$ is a translation of $\scrP_1$, and therefore
\begin{align*}
\mv(\scrP_1, \scrP_2, \scrP_3) = 3! \vol(\scrP_1) = 2
\end{align*}
If $f_j$ are polynomials such that $\np(f_j) = \scrP_j$, then
\begin{align*}
f_1 &= a_1 + b_1x_1x_2 + c_1x_2x_3 + d_1x_3x_1\\
f_2 &= a_2 + b_2x_1x_2+ c_2x_2x_3  + d_2x_3x_1 \\
f_3 &= x_3(a_3 + b_3x_1x_2 + c_3x_2x_3  + d_3x_3x_1)
\end{align*}
where $a_j, b_j, c_j, d_j \in \ktorus$. We write $V$ for the set of common zeroes of $f_1, \ldots, f_3$ on $\kk^3$, and $V^*$ for $V \cap \nktoruss{3}$.
\begin{defnlist}
\item \label{aff-non-deg-1-0} If all $a_j, b_j, c_j, d_j$ are generic, then it is straightforward to check directly that $V = V^*$. Therefore \cref{bkk-bound-thm} implies that $\multpisonodots{\scrA_1, \scrA_2, \scrA_3}{\kk^3}  = \multpisonodots{f_1, f_2, f_3}{\kk^3}  = \mv(\scrP_1, \scrP_2, \scrP_3) = 2$.
\item \label{aff-non-deg-1-1} If $a_1 = a_2$, $b_1 = b_2$, and the remaining coefficients are generic, then $V = V^* \cup C$, where $C := \{x_3= a_1 + b_1x_1x_2 = 0\}$ is a positive dimensional component of $V(f_1, f_2, f_3)$. However, $f_1, f_2, f_3$ still satisfy \eqref{b-non-degeneracy}, and \cref{bkk-non-degenerate-thm} implies that $\multpisonodots{f_1, f_2, f_3}{\kk^3} = 2 = \multpisonodots{\scrA_1, \scrA_2, \scrA_3}{\kk^3}$.
\item \label{aff-non-deg-1-2} If $a_1 = a_2 = a_3$, $b_1 = b_2 = b_3$, and the rest of the coefficients are generic, then again $V = V^* \cup C$. However, \eqref{b-non-degeneracy} fails for the weighted order $\nu$ corresponding to weights $(-1,1,2)$ for $(x,y,z)$, and \cref{bkk-non-degenerate-thm} implies that $\multpisonodots{f_1, f_2, f_3}{\kk^3} < 2 = \multpisonodots{\scrA_1, \scrA_2, \scrA_3}{\kk^3}$. (It is straightforward to verify directly that in this case $V^* = \emptyset$ and $\multpisonodots{f_1, f_2, f_3}{\kk^3} = 0$.)
\end{defnlist}
\end{example}

Part \ref{aff-non-deg-1-1} of \cref{ex-aff-non-deg-1} shows that it is possible that $V(f_1, \ldots, f_n)$ has a positive dimensional component on $\kk^n$, but still $\multfkniso = \multAkniso$ (where $\scrA_j = \supp(f_j)$). (This does not happen in the case of $\nktorus$, see \cref{positive-dimensional-prop}.) Moreover, since the intersection of the curve $C$ with the ``torus'' of the $(x_1,x_2)$-plane is nonempty, in part \ref{aff-non-deg-1-1} of \cref{ex-aff-non-deg-1}, condition \eqref{non-deg-infinity} is violated for $I = \{1,2\}$. However, note that the intersections of $\scrP_1$ and $\scrP_2$ with the $(x_1,x_2)$-plane (in $\rr^n$) are {\em dependent} in the terminology of \cref{dependent-defn}. Moreover, in part \ref{aff-non-deg-1-2} of \cref{ex-aff-non-deg-1}, where $\multpisonodots{f_1, f_2, f_3}{\kk^3} < \multpisonodots{\scrA_1, \scrA_2, \scrA_3}{\kk^3}$, condition \eqref{non-deg-infinity} is violated with $I = \{1, 2, 3\}$, and the corresponding polytopes are {\em independent}. This motivates the following definition.

\begin{defn} \label{ri-dependent-defn}
An ordered collection $\scrB = (\scrB_1, \ldots, \scrB_m)$, $m \geq 1$, of collections of finite subsets of $\rr^n$ is called {\em $\ri$-dependent} if there is a nonempty subset $J$ of $[m]$ such that $\scrB^I_j := \scrB_j \cap \ri$ is nonempty for each $j \in J$, and the collection $\{\conv(\scrB^I_j): j \in J\}$ of convex polytopes is dependent (\cref{dependent-defn}); otherwise we say that $\scrB$ is {\em $\ri$-independent}.
\end{defn}

\Cref{ex-aff-non-deg-1} suggests that
\begin{prooflist}
\item \label{non-deg-ind-infinity} Condition \eqref{non-deg-infinity} should be checked only for those $I \subseteq [n]$ such that $\scrA$ is $\ri$-independent.
\end{prooflist}
This, however, is not enough, as the next example shows.

\def\shiftx{9}

\def\colorzero{blue}
\def\opazero{0.5}
\def\viewx{70}
\def\viewy{5}
\def\titlex{2}
\def\titley{-1}
\def\scalefactor{0.6}
\def\colorzero{blue}
\def\colorone{green}

\begin{figure}[h]
\begin{center}
\begin{tikzpicture}[scale=0.6]
\pgfplotsset{every axis/.append style = {view={\viewx}{\viewy}, axis lines=middle, enlargelimits={upper}}}

\begin{scope}
\begin{axis}
	\addplot3 [fill=blue,opacity=\opazero, thick] coordinates{(0,1,1) (1,0,0) (1,0,1)};
	\addplot3 [fill=red,opacity=\opazero, thick] coordinates{(0,0,0) (1,0,0) (1,0,1)};
	\addplot3 [thick, dashed] coordinates{(0,0,0) (0,1,1)};

\end{axis}
\draw (\titlex,\titley) node {$\scrP_1 = \scrP_2$};
\end{scope}

\begin{scope}[shift={(\shiftx,0)}]
\begin{axis}
	\addplot3 [fill=red,opacity=\opazero, thick] coordinates{(0,0,1) (0,1,1) (1,0,2)};
	\addplot3 [fill=blue,opacity=\opazero, thick] coordinates{(0,1,1) (1,0,2) (0,1,2)};
	\addplot3 [thick, dashed] coordinates{(0,0,1) (0,1,2)};

\end{axis}
\draw (\titlex,\titley) node {$\scrP_3$};
\end{scope}

\def\factorone{1}
\def\factortwo{1}
\def\factorthree{1}
\def\factorfour{1}
\begin{scope}[shift={(2*\shiftone,0)}]
\begin{axis}[hide axis, enlargelimits={0.15}]
	\addplot3 [ultra thick, red, ->] coordinates{(0,0,0) (0,-\factorone,0)};
	\addplot3 [ultra thick, blue, ->] coordinates{(0,0,0) (\factortwo, \factortwo,0)};
	\addplot3 [ultra thick, ->] coordinates{(0,0,0) (-\factorthree,-\factorthree,\factorthree)};
	\addplot3 [ultra thick, ->] coordinates{(0,0,0) (0,\factorfour,-\factorfour)};

	\draw (axis cs:0,-\factorone,0) node [below] {(0,-1,0)};
	\draw (axis cs: \factortwo,\factortwo,0) node [above] {(1,1,0)};
	\draw (axis cs:-\factorthree,-\factorthree,\factorthree) node [right] {(-1,-1,1)};
	\draw (axis cs: 0,\factorfour,-\factorfour) node [below] {(0,1,-1)};
\end{axis}

\draw (\titlex,\titley) node [right, text width=3.5cm] {outer normals to facets of $\scrP_1$};
\end{scope}
\end{tikzpicture}
\end{center}
\caption{Newton polytopes of \cref{ex-aff-non-deg-2}} \label{fig-ex-aff-non-deg-2}
\end{figure}

\begin{example}\label{ex-aff-non-deg-2}
Consider the following system of polynomials:
\begin{align*}
f_1 &= a_1 + b_1x_1 + c_1x_2x_3 + d_1x_3x_1\\
f_2 &= a_2 + b_2x_1 + c_2x_2x_3  + d_2x_3x_1 \\
f_3 &= x_3(a_3 + b_3x_2 + c_3x_2x_3  + d_3x_3x_1)
\end{align*}
where $a_j,b_j, c_j, d_j \in \ktorus$, and $\scrA_j  := \supp(f_j)$, $j = 1, 2, 3$. We continue to write $V := V(f_1, f_2, f_3)$ and $V^* := V \cap \nktoruss{3}$. When all the coefficients are generic, then it is straightforward to check that $V = V^*$, so that \cref{bkk-bound-thm} implies that
\begin{align*}
\multpisonodots{f_1, f_2, f_3}{\kk^3}
	&= \multpisonodots{\scrA_1, \scrA_2, \scrA_3}{\kk^3}
	= \mv(\scrP_1, \scrP_2, \scrP_3)
\end{align*}
We compute $\mv(\scrP_1, \scrP_2, \scrP_3)$ using \cref{rational-mv'-prop}. If $\nu$ is the primitive outer normal to any of the facets of $\scrP_1 = \scrP_2$, it is straightforward to check that the image of the corresponding facet under the map $\psi_\nu$ from the definition of $\mv'_\nu(\cdot, \ldots, \cdot)$ is (up to a translation) the triangle $\scrT$ with vertices $(0,0),(0,1),(1,0)$, and therefore $\mv'_\nu(\ld_\nu(\scrP_1), \ld_\nu(\scrP_2)) = 2! \area(\scrT) = 1$.
It follows that
\begin{align*}
\multpisonodots{\scrA_1, \scrA_2, \scrA_3}{\kk^3}
	&= \mv(\scrP_1, \scrP_2, \scrP_3) \\
	&= \max_{\scrP_3}(0,-1,0) + \max_{\scrP_3}(0,1,-1) + \max_{\scrP_3}(1,1,0) + \max_{\scrP_3}(-1,-1,1) \\
	&= 0 + 0 + 1+ 1 \\
	& = 2
\end{align*}
If $a_1 = a_2$ and $b_1 = b_2$ and the other coefficients are generic, then it is straightforward to check directly that $V^* = \emptyset$ and $V$ is the curve $\{x_3 = a_1+b_1x_1 = 0\}$, so that $\multpisonodots{f_1, f_2, f_3}{\kk^3} = 0$. However, the only $I \subseteq \{1, 2,3\}$ such that $\scrA$ is $\ri$-independent is $I = \{1, 2, 3\}$, and it is straightforward to check that $f_j|_{\Ki} = f_j$ are in fact $\scrA$-non-degenerate at infinity. In particular, $f_1, f_2, f_3$ satisfy condition \ref{non-deg-ind-infinity}, but $\multpisonodots{f_1, f_2, f_3}{\kk^3} < \multpisonodots{\scrA_1, \scrA_2, \scrA_3}{\kk^3} $.
\end{example}

Given $I' \subseteq I \subseteq [n]$ and a weighted order $\nu$ on $\kk[x_i: i \in I]$, we say that $\nu$ is centered at $\Kstariprime$ if $\nu(x_i) = 0$ for all $i \in I'$ and $\nu(x_i) > 0$ for all $i \in I\setminus I'$. It is straightforward to check that in \cref{ex-aff-non-deg-2}, the only nonzero weighted orders $\nu$ such that $\In_\nu(f_j)$, $j = 1, 2, 3$, have a common zero on $\nktoruss{3}$ are of the form $(0, 0, \epsilon)$ for $\epsilon > 0$, i.e.\ they are centered at $\Kstariprime$ with $I' := \{1,2\}$. It turns out that $\scrA$ is {\em hereditarily $\riprime$-dependent} (see \cref{non-deg-section}) and therefore \cref{ex-aff-non-deg-2} suggests that for $\multfkniso$ to be equal to $\multAkniso$, the following condition needs to be satisfied in addition to \ref{non-deg-ind-infinity}:
\begin{prooflist}[resume]
\item \label{non-deg-ind-finity} For each $I \subseteq [n]$ such that $\scrA$ is {\em not} hereditarily $\ri$-dependent, $\In_{\Ai_j, \nu}(f_j|_{\Ki})$ do not have any common zero on $\nktorus$ for all weighted orders $\nu$ on $\kk[x_i: i \in I]$ which are centered at $\Kstariprime$ for some $I' \subsetneq I$ such that $\scrA$ is hereditarily $\riprime$-dependent.
\end{prooflist}

\begin{example}\label{ex-aff-non-deg-3}
Let $f_1, \ldots, f_4$ be polynomials in $(x_1, \ldots, x_4)$ such that $f_1$ and $f_2$ are polynomials in $(x_1, x_2)$ with nonzero constant terms, the (two dimensional) mixed volume of $\np(f_1)$ and $\np(f_2)$ is nonzero, and
\begin{align*}
f_j &= x_3f_{j,1}(x_1, x_2) + x_4f_{j,2}(x_1,x_2) \in \kk[x_1, x_2, x_3, x_4],\ j = 3, 4,
\end{align*}
where $f_{3,1}, f_{3,2}, f_{4,1}, f_{4,2}$ are nonzero polynomials in $(x,y)$ such that $\np(f_{3,1})$ and $\np(f_{4,1})$ are positive dimensional. Let $\scrA_j$ be the support of $f_j$ and $\scrP_j$ be the Newton polytope of $f_j$, $j = 1, \ldots, 4$. The only $I \subseteq \{1, 2, 3, 4\}$ such that the supports of $f_j$ are $\ri$-independent is $I = \{1,2\}$. It follows that for generic coefficients, all the common zeroes of $f_1, \ldots, f_4$ are isolated and contained in the $(x_1,x_2)$-plane. Moreover, \cref{extended-bkk-bound-0} implies that
\begin{align*}
\multpisonodots{\scrA_1, \scrA_2, \scrA_3, \scrA_4}{\kk^4}
	= 	\mv(\scrP^{\{1,2\}}_1, \scrP^{\{1,2\}}_2) \times
		\multpnodots{\pi_{\{3,4\}}(\scrP_3),\pi_{\{3,4\}}(\scrP_4)}{\origin}
	= \mv(\scrP_1, \scrP_2)
\end{align*}
Now fix BKK non-degenerate $f_1, f_2$, and a common zero $z = (z_1, z_2)$ of $f_1,f_2$ on $\nktoruss{2}$. Take $f_{3,1}$ and $f_{4,1}$ such that $f_{3,1}(z) = f_{4,1}(z) = 0$. Then $\{(z_1,z_2, t, 0): t \in \kk\} \subseteq  V(f_1, f_2, f_3, f_4)$, so that $(z_1, z_2, 0,0)$ is no longer an isolated point of $V(f_1, f_2, f_3, f_4)$ (even though it is an isolated point of $V(f_1, f_2, f_3, f_4) \cap \Kii{\{1,2\}}$). It follows that $\multpisonodots{f_1, f_2,f_3, f_4}{\kk^4} <  \mv(\scrP_1, \scrP_2) = \multpisonodots{\scrA_1, \scrA_2, \scrA_3, \scrA_4}{\kk^4}$. However, $f_1, \ldots, f_4$ satisfy both conditions \ref{non-deg-ind-infinity} and \ref{non-deg-ind-finity}.
\end{example}

\Cref{ex-aff-non-deg-3} leads us to another condition that needs to be satisfied in addition to \ref{non-deg-ind-infinity} and \ref{non-deg-ind-finity} for $\multfkniso$ to be equal to $\multAkniso$:
\begin{prooflist}[resume]
\item \label{non-deg-dep} For each $I \subseteq [n]$ such that $\scrA$ is hereditarily $\ri$-dependent, $\In_{\Ai_j, \nu}(f_j|_{\Ki})$ do not have any common zero on $\nktorus$ for all weighted orders $\nu$ on $\kk[x_i: i \in I]$ which are centered at $\Kstariprime$ for some $I' \subsetneq I$ such that $\scrA$ is not hereditarily $\riprime$-dependent.
\end{prooflist}
Note that condition \ref{non-deg-dep} is precisely the opposite of condition \ref{non-deg-ind-finity}.

\section{Non-degeneracy conditions} \label{non-deg-section}
Let $\scrA := (\scrA_1, \ldots, \scrA_n)$ be a collection of finite subsets of $\znzero$ and $I \subseteq [n]$. We say that $\scrA$ is {\em hereditarily $\ri$-dependent} if $\scrA$ is $\ri$-dependent (see \cref{ri-dependent-defn}) and there is $I' \supseteq I$ such that
\begin{defnlist}
\item $\scrA$ is $\riprime$-dependent,
\item $|\tiprimeA| = |I'|$,
\item $|\titildeA| > |\tilde I|$ for each $\tilde I$ such that $I \subseteq \tilde I \subsetneq I'$.
\end{defnlist}

\begin{rem} \label{empty-remark}
If $I = \emptyset$, then $\ri$ is the origin, and $\scrA$ is $\ri$-dependent if and only if the origin belongs to some $\scrA_j$. However, even if $\scrA$ is $\ri$-dependent, it might not be hereditarily $\ri$-dependent, see \cref{empty-figure}.
\end{rem}

\begin{figure}[h]
\begin{center}

\begin{tikzpicture}[scale=0.45]
\def\shiftone{7.5}

\def\opazero{0.5}
\def\gridx{4.5}

\def\gridy{4.5}
\def\colorone{red}

\draw [gray,  line width=0pt] (-0.5,-0.5) grid (\gridx,\gridy);
\draw [<->] (0, \gridy) |- (\gridx, 0);
\draw[ultra thick, \colorone]  (3,0) -- (0,3) -- (0,0) -- cycle;
\fill[\colorzero, opacity=\opazero ] (3,0) -- (0,3) -- (0,0) -- cycle;
\def\tx{0.5}
\def\ty{0.5}
\draw (\tx,\ty) node {\picfontsize $\scrA_1$};

\begin{scope}[shift={(\shiftone,0)}]
	\draw [gray,  line width=0pt] (-0.5,-0.5) grid (\gridx,\gridy);
	\draw [<->] (0, \gridy) |- (\gridx, 0);
	\draw[ultra thick, \colorone]  (1,1) --  (4,1) -- (4,4) -- (1,4) -- cycle;
	\fill[\colorzero, opacity=\opazero ] (1,1) --  (4,1) -- (4,4) -- (1,4) -- cycle;
	\def\tx{2.5}
	\def\ty{2.5}
	\draw (\tx,\ty) node {\picfontsize $\scrA_2$};
\end{scope}

\begin{scope}[shift={(2*\shiftone,0)}]
	\draw [gray,  line width=0pt] (-0.5,-0.5) grid (\gridx,\gridy);
	\draw [<->] (0, \gridy) |- (\gridx, 0);
	\draw[ultra thick, \colorone]  (4,0) -- (1,4) -- (0,0) -- cycle;
	\fill[\colorzero, opacity=\opazero ] (4,0) -- (1,4) -- (0,0) -- cycle;
	\def\tx{1.5}
	\def\ty{1.5}
	\draw (\tx,\ty) node {\picfontsize $\scrA_3$};
\end{scope}
\end{tikzpicture}

\end{center}
\caption{Both $(\scrA_1, \scrA_2)$ and $(\scrA_2, \scrA_3)$ are $\rr^I$-dependent for $I = \emptyset$. However, only the latter pair is hereditarily $\rr^I$-dependent (with $I' = \{2\}$).} \label{empty-figure}
\end{figure}

The relevance of ``hereditary dependence'' to affine B\'ezout problem is given by \cref{hereditary-prop} below: it states that if $\scrA$ is hereditarily $\ri$-dependent, then for all $(f_1, \ldots, f_n) \in \scrL(\scrA)$, either $V^I := V(f_1, \ldots, f_n) \cap \Kstari$ is empty, or all points of $V^I$ are non-isolated in (the possibly larger set) $V(f_1, \ldots, f_n)$ (however, points of $V^I$ might be isolated in $V^I$ itself!). This is not necessarily true if $\scrA$ is simply $\ri$-dependent, e.g.\ the system $(x + y - 1, 2x - y - 2)$ (over a field of characteristic not equal to two) has an isolated zero on the coordinate subspace $y = 0$.

\begin{prop} \label{hereditary-prop}
If $\scrA$ is hereditarily $\ri$-dependent, then $V(f_1, \ldots, f_n)$ has no isolated point on $\Kstari$ for each $(f_1, \ldots, f_n) \in \scrL(\scrA)$.
\end{prop}

We prove \cref{hereditary-prop} in \cref{reducsection}. Now we introduce the correct non-degeneracy condition for (arbitrary open subsets of) the affine space. Let $U$ be a Zariski open subset of $\kk^n$. Let $\excA$ and $\excU$ be as in \cref{first-section}. Define
\begin{align}
\dUA &:= \{I \subseteq [n]: I \not \in \excU \cup \excA,\ \scrA\ \text{is hereditarily $\ri$-dependent}\}  \label{dUA}\\
\iUA &:= \{I \subseteq [n]: I \not \in \excU \cup \excA,\ \scrA\ \text{is not hereditarily $\ri$-dependent}\}  \label{iUA}
\end{align}
We say that polynomials $f_1, \ldots, f_n$ are {\em $(U,\scrA)$-non-degenerate} if the following conditions are true:
\begin{defnlist}
\item \label{U-property} For each nonempty $I \in \iUA$, there is no common zero of $f_1|_{\Ki}, \ldots, f_n|_{\Ki}$ on $\Kstari \setminus U$ (note that this condition is vacuously true when $U = \kk^n$, or more generally, when $U = \Kns$ (see \eqref{Kns}) for some collection $\mscrS$ of subsets of $[n]$),
\item \label{iUA-property} For each nonempty $I \in \iUA$ and for each weighted order $\nu$ on $\kk[x_i: i \in I]$ such that $\nu$ is centered at infinity or at $\Kstariprime$ for some $I' \in \excU \cup \excA \cup \dUA$, there is no common zero of $\In_{\Ai_1, \nu}(f_1|_{\Ki}), \ldots, \In_{\Ai_n,\nu}(f_n|_{\Ki})$ on $\nktorus$.
\item \label{dUA-property}  For each nonempty $I \in \dUA$ and for each weighted order $\nu$ on $\kk[x_i: i \in I]$ such that $\nu$ is centered at $\Kstariprime$ for some $I' \in \iUA$, there is no common zero of $\In_{\Ai_1, \nu}(f_1|_{\Ki}), \ldots, \In_{\Ai_n,\nu}(f_n|_{\Ki})$ on $\nktorus$.
\end{defnlist}

We prove the following result in \cref{non-deg-1-proof-section}.

\begin{thm}[{\cite{toricstein}}] \label{non-deg-1}
Assume $\multAiso{U} > 0$. Then for each $(f_1, \ldots, f_n) \in \scrL(\scrA)$, the following are equivalent:
\begin{enumerate}
\item \label{non-deg-1-0} $\multfiso{U} = \multAiso{U}$,
\item \label{non-deg-1-1}  $f_1, \ldots, f_n$ are $(U,\scrA)$-non-degenerate.
\end{enumerate}
The collection $\scrN(U, \scrA)$ of all $(U, \scrA)$-non-degenerate $(f_1, \ldots, f_n) \in \scrL(\scrA)$ is a nonempty Zariski open subset of $\scrL(\scrA)$.

\end{thm}

In both \cref{ex-aff-non-deg-1,ex-aff-non-deg-2}, $\dUUA{\kk^3}$ is the set $\{\emptyset, \{1\}, \{2\}, \{1,2\}\}$ of all subsets of $\{1,2\}$, and $\iUUA{\kk^3}$ is the set of remaining subsets of $\{1,2,3\}$. It is straightforward to check that in cases \ref{aff-non-deg-1-0} and \ref{aff-non-deg-1-1} of \cref{ex-aff-non-deg-1}, $f_1, f_2, f_3$ are $(\kk^3,\scrA)$-non-degenerate, but in case \ref{aff-non-deg-1-2}, condition \ref{iUA-property} of $(\kk^3,\scrA)$-non-degeneracy fails for $I = \{1, 2, 3\}$ and $\nu = (-1,1,2)$, which is centered at infinity. In \cref{ex-aff-non-deg-2}, condition \ref{iUA-property} of $(\kk^3,\scrA)$-non-degeneracy fails for $I = \{1, 2, 3\}$ and $\nu = (0,0,1)$, which is centered at $\Kstariprime$, where $I' := \{1,2\}$. In the scenario of \cref{ex-aff-non-deg-3}, $\iUUA{\kk^4}$ is the collection of all subsets of $\{1,2\}$, and $\dUUA{\kk^4}$ is the set of remaining subsets of $\{1,2,3,4\}$, and condition \ref{dUA-property} of $(\kk^4,\scrA)$-non-degeneracy fails for $I = \{1, 2, 3,4\}$ and $\nu = (0,0,1,2)$, which is centered at $\Kstariprime$, where $I' := \{1,2\}$.

\subsection{A more efficient formulation of $(U,\scrA)$-non-degeneracy}
In this section we describe a criterion equivalent to $(U,\scrA)$-non-degeneracy, but which involves checking fewer conditions. Recall that for $I \subseteq [n]$, we write $\tiA := \{j \in [n]:  \scrA_j \cap \ri \neq \emptyset\}$. Define
\begin{align}
\begin{split}
\dUAstar
	&:= \{I\in \dUA: I \neq \emptyset,\ |\tiA| = |I|\} \\
	&= \{I \subseteq [n]:
		I \neq \emptyset,\ |\tiA| = |I|,\ \text{$\scrA$ is $\ri$-dependent} \}
\end{split} \label{dUAstar} \\
\begin{split}
\iUAstar
	&:= \{I \in \iUA:  I \neq\emptyset,\ |\tiA| = |I|\} \\
	&= \{I \subseteq [n]:
		I \neq \emptyset,\ |\tiA| = |I|,\ \text{$\scrA$ is $\ri$-independent} \}
\end{split} \label{iUAstar}
\end{align}

\begin{prop}[{\cite{toricstein}}] \label{non-deg-2}
For polynomials $f_1, \ldots, f_n$ in $(x_1, \ldots, x_n)$, the following are equivalent:
\begin{enumerate}
\item $f_i$'s are $(U, \scrA)$-non-degenerate.
\item
\begin{enumerate}[label=(\roman{enumii})]
\item property \ref{U-property} of $(U, \scrA)$-non-degeneracy holds,
\item property \ref{iUA-property} of $(U, \scrA)$-non-degeneracy holds with $\iUA$ replaced by $\iUAstar$,
\item property \ref{dUA-property} of $(U, \scrA)$-non-degeneracy holds with $\dUA$ replaced by $\dUAstar$.
\end{enumerate}
\end{enumerate}
\end{prop}

We prove \cref{non-deg-2} in \cref{reducsection}.

\begin{example}[Warning!]
In property \ref{iUA-property} of $(U, \scrA)$-non-degeneracy $\dUA$ can {\em not} be replaced by $\dUAstar$, and in property \ref{dUA-property} of $(U, \scrA)$-non-degeneracy $\iUA$ can {\em not} be replaced by $\iUAstar$. Indeed, at first consider the system
\begin{align*}
f_1 &= 1 + x_1\\
f_2 &= 1 + x_1 + x_2\\
f_3 &= 1 + x_1 + 2x_2 + ax_3x_4\\
f_4 &= x_4(1 +  x_1 + bx_2 + cx_3 + dx_4)
\end{align*}
for generic $a, b, c, d \in \ktorus$. Let $U = \kk^4$ and $\scrA_j =  \supp(f_j)$, $j = 1, \ldots, 4$. Then $\dUA$ is the collection of all subsets of $\{1,2,3\}$, $\iUA$ is the collection of all subsets of $\{1, 2, 3, 4\}$ containing $4$, $\dUAstar = \{\{1,2,3\}\}$, and $\iUAstar = \{\{1,2,3,4\}\}$. Condition \ref{iUA-property} of $(U, \scrA)$-non-degeneracy fails with $I = \{1, 2,3,4\}$ and $\nu = (0,1,1,1)$, so that the center of $\nu$ is $\Kstarii{I'}$, where $I' := \{1\}\in \dUA$. However, it is straightforward to check that condition \ref{iUA-property} would not have been violated had $\dUA$ been replaced by $\dUAstar$. Now consider the system
\begin{align*}
g_1 &= 1 + x_1 + x_2 + x_3 \\
g_2 &= 1 + x_1 + 2x_2 + 3x_3 \\
g_3 &= x_2(ax_1 + bx_2 + cx_3)+ x_3(a'x_1+b'x_2+c'x_3) \\
g_4 &= x_4(1+x_1)
\end{align*}
where $a,b,c,a',b',c'$ are generic elements in $\ktorus$. Let $\scrB := (\supp(g_1), \ldots, \supp(g_4))$, and $U := \kk^4$. Then $\dUB$ is the collection of all subsets of $\{1,2,3,4\}$ containing $4$, $\iUB$ is the collection of all subsets of $\{1,2,3,4\}$ not containing $4$, $\dUBstar = \{\{1,2,3,4\}\}$, and $\iUBstar = \{\{1,2,3\}\}$. It is straightforward to check that $g_1, g_2, g_3, g_4$ violate condition \ref{dUA-property} of $(U, \scrB)$-non-degeneracy with $I = \{1, 2,3,4\}$ and $\nu = (0,1,1,1)$ (so that the center of $\nu$ is $\Kstarii{I'}$, where $I' := \{1\}\in \iUB$). It is also straightforward to check that condition \ref{dUA-property} would not have been violated had $\iUB$ been replaced by $\iUBstar$.
\end{example}

The following combinatorial description of strict monotonicity of $\multpiso{\cdot}{\cdot}{U}$ follows from \cref{non-deg-1,non-deg-2} exactly as in the proof of \cref{strictly-monotone-0}.

\begin{cor}
Let $\scrB_j \subseteq \conv(\scrA_j)$, $j = 1, \ldots, n$. Then $\multAiso{U} \geq \multBiso{U}$. If $\multAiso{U} > 0$, then the following are equivalent:
\begin{enumerate}
\item $\multAiso{U} = \multBiso{U}$,
\item
\begin{enumerate}
\item For each nonempty $I \in \iUA$ and for each weighted order $\nu$ on $\kk[x_i: i \in I]$ such that $\nu$ is centered at infinity or at $\Kstariprime$ for some $I' \in \excU \cup \excA \cup \dUA$, the collection $\{\In_\nu(\nd(\Bi_j)): \In_\nu(\nd(\Ai_j)) \cap \supp(\scrB_j) \neq \emptyset\}$ of polytopes is dependent, and
\item for each nonempty $I \in \dUA$ and for each weighted order $\nu$ on $\kk[x_i: i \in I]$ such that $\nu$ is centered at  $\Kstariprime$ for some $I' \in \iUA$, the collection $\{\In_\nu(\nd(\Bi_j)): \In_\nu(\nd(\Ai_j)) \cap \supp(\scrB_j) \neq \emptyset\}$ of polytopes is dependent.
\end{enumerate}
\item
\begin{enumerate}
\item For each nonempty $I \in \iUAstar$ and for each weighted order $\nu$ on $\kk[x_i: i \in I]$ such that $\nu$ is centered at infinity or at $\Kstariprime$ for some $I' \in \excU \cup \excA \cup \dUA$, the collection $\{\In_\nu(\nd(\Bi_j)): \In_\nu(\nd(\Ai_j)) \cap \supp(\scrB_j) \neq \emptyset\}$ of polytopes is dependent, and
\item for each nonempty $I \in \dUAstar$ and for each weighted order $\nu$ on $\kk[x_i: i \in I]$ such that $\nu$ is centered at  $\Kstariprime$ for some $I' \in \iUA$, the collection $\{\In_\nu(\nd(\Bi_j)): \In_\nu(\nd(\Ai_j)) \cap \supp(\scrB_j) \neq \emptyset\}$ of polytopes is dependent. \qed
\end{enumerate}
\end{enumerate}
\end{cor}

\section{Proof of the non-degeneracy conditions} \label{non-deg-proof-section}

\subsection{Reduction to the more efficient non-degeneracy criterion} \label{reducsection}
In this section we prove \cref{hereditary-prop,non-deg-2}.

\begin{proof}[Proof of \cref{hereditary-prop}]
Let $I' \supseteq I$ be as in the definition of hereditary non-degeneracy. By restricting all $f_j$'s to $\Kiprime$, we may assume that $I' = [n]$. Let $Z^I$ be the set of isolated points $V(f_1, \ldots, f_n)$ which are on $\Kstari$. Assume to the contrary of the claim that $Z^I \neq \emptyset$. Let $(g_1, \ldots, g_n) \in \scrL(\scrA)$ be a system such that
\begin{enumerate}
\item $\np(g_j) = \conv(\scrA_j)$, $j = 1, \ldots, n$, and
\item $g_1|_{\Kitilde}, \ldots, g_n|_{\Kitilde}$ are {\em properly $\Atildei$-non-degenerate} (see \cref{properly-non-degenerate-section}) for each $\tilde I \subseteq [n]$, where $\Atildei := (\scrA_1 \cap \rii{\tilde I}, \ldots, \scrA_n \cap \rii{\tilde n})$ (\cref{properly-non-degenerate-claim} implies that the set of such systems is a nonempty Zariski open subset of $\scrL(\scrA)$).
\end{enumerate}
Let $t$ be a new indeterminate, and $h_j := tg_j + (1-t)f_j \in \kk[x_1, \ldots, x_n, t]$, $j = 1, \ldots, n$. Pick $z \in Z^I$. Then $(z,0)$ is an isolated point of $V(h_1, \ldots, h_n, t)$ on $\kk^{n+1}$, and therefore \cref{thm:pure-dimension} implies that there is a Zariski open neighborhood $U$ of $(z,0)$ in $\kk^{n+1}$ such that $C := V(h_1, \ldots, h_n) \cap U$ is a curve. Let $C'$ be an irreducible component of $C$ containing $(z,0)$, and $\bar C'$ be the closure of $C'$ in $\pp^n \times \pp^1$. Then $C' \not\subseteq \kk^n \times \{\epsilon\}$ for any $\epsilon \in \kk$, and \cref{linear-intersection} implies that $\bar C'$ intersects $\pp^n \times \{1\}$. Pick a branch $B'$ of $\bar C'$ centered at a point $(z',1) \in \bar C' \cap (\pp^n \times \{1\})$. Define $I_B \subseteq [n+1]$ and the weighted order $\nu_B$ on the coordinate ring of $\Kib$ as in \cref{IB-defn}. Let $\tilde I := I_B \cap [n]$ and $\tilde \nu$ be the restriction of $\nu_B$ to $\kk[x_i : i \in \tilde I]$. Since $\np(f_j) \subseteq \np(g_j)$ for each $j$, it follows that $\tilde \nu(f_j|_{\Kitilde}) \geq \tilde \nu(g_j|_{\Kitilde})$. Since $\nu_B(t-1) > 0$, it follows that $\nu_B(t) = 0$, and therefore $\In_{\nu_B}(h_j|_{\Kib}) = t\In_{\tilde \nu}(g_j|_{\Kitilde})$ for each $j$. \Cref{branch-lemma-IB} then implies that $\In_{\tilde \nu}(g_j|_{\Kitilde})$ have a common zero in $\Kstaritilde$. On the other hand, since $\bar C'$ contains the point $(z,0) \in \Kstari$, it follows that $I \subseteq \tilde I$, and therefore it follows from the definition of hereditary non-degeneracy (applied with $I' = [n]$) that either $|\titildeA| > |\tilde I|$, or $|\titildeA| =|\tilde I|$ and $\scrA$ is $\ri$-dependent. In any event, there is a nonempty subset $J$ of $\titildeA$ such that $\dim(\sum_{j \in J}\np(g_j|_{\Kitilde})) < |J|$. Since $g_1|_{\Kitilde}, \ldots, g_n|_{\Kitilde}$ are properly $\Atildei$-non-degenerate, it then follows by definition of proper non-degeneracy that  there is no common zero of $\In_{\tilde \nu}(g_j|_{\Kitilde})$ on $\Kstaritilde$. This contradiction completes the proof.
\end{proof}

\begin{proof}[Proof of \cref{non-deg-2}]
Since $\dUAstar \subseteq \dUA$ and $\iUAstar \subseteq \iUA$, it suffices to show the following:
\begin{enumerate}

\item property \ref{iUA-property} of $(U, \scrA)$-non-degeneracy holds if it holds with $\iUA$ replaced by $\iUAstar$,
\item property \ref{dUA-property} of $(U, \scrA)$-non-degeneracy holds if it holds with $\dUA$ replaced by $\dUAstar$.
\end{enumerate}
It follows from the definition of hereditary non-degeneracy that for every $I \in \iUA \setminus \iUAstar$, there exists $I' \in \iUAstar$ such that $I \subsetneq I'$ and $|\tiprimeA| > |I'|$ for each $I'$ such that $I \subseteq I' \subsetneq \tilde I$. The same statement also holds if we replace $\iUA$ and $\iUAstar$ respectively by $\dUA$ and $\dUAstar$. Since restricting all $f_j$ to $\Kitilde$ yields a system with the same number of nonzero polynomials as the number of variables, it suffices to prove the following claim: ``if there is $I \subseteq [n]$ such that $|\tiA| > |I|$ and $\In_{\Ai_1,\nu}(f_1|_{\Ki}),  \ldots, \In_{\Ai_n, \nu}(f_n|_{\Ki})$ have a common zero on $\nktorus$ for some weighted order $\nu$ on $\kk[x_i: i \in I]$, then there is $\tilde I \supsetneq I$ and a weighted order $\tilde \nu$ on $\kk[x_{\tilde i}: \tilde i \in \tilde I]$ such that $\tilde \nu$ is compatible with $\nu$, and $\In_{\Atildei_1, \tilde \nu}(f_1|_{\Kitilde}), \ldots, \In_{\Atildei_n, \tilde \nu}(f_n|_{\Kitilde})$ have a common zero in $\tilde u \in \nktorus$.'' This follows from \cref{reduction-lemma}.
\end{proof}

\subsection{Understanding condition \ref{dUA-property} of $(U, \scrA)$-non-degeneracy}
In this section we show that if condition \ref{dUA-property} of $(U, \scrA)$-non-degeneracy is violated for $(f_1, \ldots, f_n) \in \scrL(\scrA)$ with $I$, $\nu$ and $I'$, then the common zero on $\nktorus$ of $\In_{\Ai_1, \nu}(f_1|_{\Ki}), \ldots, \In_{\Ai_n, \nu}(f_1|_{\Ki})$ corresponds to a non-isolated point of $V(f_1, \ldots, f_n)$. Recall the definition of $\pi_I: \kk^n \to \Ki$ from \eqref{pi_I}.

\begin{lemma} \label{trivial-lemma}
Let $(f_1, \ldots, f_n) \in \scrL(\scrA)$, and $\nu$ be a weighted order on $\kk[x_1, \ldots, x_n]$ centered at $\Kstari$, $I \subseteq [n]$.
Assume
\begin{enumerate}
\item \label{trivially-dependent} $\conv(\scrA_1), \ldots, \conv(\scrA_n)$ are dependent polytopes in $\rr^n$, and
\item \label{trivially-common} $\In_{\scrA_1, \nu}(f_1),  \ldots,\In_{\scrA_n, \nu}(f_n)$ have a common zero $a \in \nktorus$.
\end{enumerate}
Then $\pi_{I}(a) \in \Kstari$ is a non-isolated point of the set $V(f_1, \ldots, f_n)$ of common zeroes of $f_1, \ldots, f_n$ on $\kk^n$.
\end{lemma}

Note that both conditions \eqref{trivially-dependent} and \eqref{trivially-common} are necessary for the conclusion of \cref{trivial-lemma} to hold. For example, $f_1 = 1 + x_1, f_2 = 1 + x_1 + x_2$ satisfy condition \eqref{trivially-common}, but not \eqref{trivially-dependent}, with $I = \{1\}$, $\nu = (0,1)$, $\scrA_j = \supp(f_j)$, $j = 1,2$, and $a = (-1,c)$ for some arbitrary $c \in \kk^*$. In this case $\pi_I(a) = (-1,0)$ is an {\em isolated} point of $V(f_1, f_2)$. On the other hand, $f_1 = 1 + x_1, f_2 = x_2$, and $\scrA_j = \supp(f_j)$, $j = 1,2$, satisfy, with the same $I$, $\nu$ and $a$, condition \eqref{trivially-dependent}, but not \eqref{trivially-common}, and $\pi_I(a) = (-1,0)$ is again an isolated point of $V(f_1, f_2)$.

\begin{proof}[Proof of \cref{trivial-lemma}]
It is immediate to check that $\pi_{I}(a)$ is in $V(f_1, \ldots, f_n)$; we only have to show that it is non-isolated in there. Since $\scrP_j := \conv(\scrA_j)$, $j = 1, \ldots, n$, are dependent, it follows that there is $J \subseteq [n]$ such that  $p:= \dim(\sum_{j \in J} \scrP_j) <|J|$. Let $\Pi$ be the (unique) $p$-dimensional linear subspace of $\rr^n$ such that $\sum_{j \in J} \scrP_j$ is contained in a translate of $\Pi$. Let $\nu_j := \nu(x_j)$, $j = 1, \ldots, n$. We identify $\nu$ with the element of $\rnstar$ with coordinates $(\nu_1, \ldots, \nu_n)$ with respect to the basis dual to the standard basis of $\rr^n$. Let $\Pi_0 := \{\alpha \in \Pi: \langle \nu, \alpha \rangle = 0\}$ and $r := \dim(\Pi_0)$. Choose a basis $\alpha_1, \ldots, \alpha_r$ of $\Pi_0 \cap \zz^n$. Let $c_i := a^{\alpha_i}$, $i = 1, \ldots, n$. Let $Y$ be the subvariety of $\nktorus$ determined by $x^{\alpha_i} - c_i$, $i = 1, \ldots, r$, and $\bar Y$ be the closure of $Y$ in $\kk^n$. Then $Y$, and therefore $\bar Y$, is an irreducible variety of codimension $r$ in $\kk^n$.

\begin{proclaim} \label{pi_I(a)-claim}
$\pi_I(a) \in \bar Y$. Moreover, if $g$ is any Laurent polynomial in $(x_1, \ldots, x_n)$ such that $\supp(g) \subseteq \Pi_0$, then $g$ restricts to a constant function on $Y$ with value $g(a)$.
\end{proclaim}

\begin{proof}
The second assertion is obvious, so we prove the first assertion. If $\nu$ is the trivial weighted order, then $\pi_I$ is the identity and therefore $\pi_I(a) = a \in Y$. Otherwise let $C$ be the curve on $\kk^n$ parametrized by $t \mapsto c(t) := (a_1t^{\nu_1}, \ldots, a_nt^{\nu_n})$. Then $C \cap \nktorus \subseteq Y$, so that $C \subseteq \bar Y$. Since $\pi_I(a) = c(0) \in C$, the claim is proved.
\end{proof}

Note that $r$ equals either $p$ or $p -1$. If $r = p$, then $\Pi_0 = \Pi$ and for each $j \in J$, $f_j$ is $\nu$-homogeneous and is of the form $f_j = x^{\beta_j}g_j$ for some $\beta_j \in \zz^n$ and Laurent polynomial $g_j$ such that $\supp(g_j) \subseteq \Pi_0$. \Cref{pi_I(a)-claim} implies that $f_j|_Y \equiv 0$ for each $j \in J$, so that $\pi_I(a) \in \bar Y \subseteq V(f_j :j \in J)$. It follows that at least one of the irreducible components of $V(f_j :j \in J)$ containing $\pi_I(a)$ has codimension smaller than $|J|$ in $\kk^n$. The lemma then follows due to \cref{thm:pure-dimension}. It remains to consider the case that $r = p-1$. \Cref{basis-lemma} implies that $\alpha_1, \ldots, \alpha_r$ can be extended to a basis $\alpha_1, \ldots, \alpha_n$ of $\zz^n$ such that $\alpha_1, \ldots, \alpha_{r+1}$ is a basis of $\Pi$, and $\langle \nu, \alpha_j \rangle \geq 0$ for each $j = 1, \ldots, n$; this in particular implies that $\langle \nu, \alpha_{r+1} \rangle > 0$. Let $y_i := x^{\alpha_i}$, $i = 1, \ldots, n$. Then the $y_i$ form a system of coordinates on $\nktorus$ and the projection onto $(y_{r+1}, \ldots, y_n)$ restricts to an isomorphism $Y \cong \nktoruss{n-r}$. Pick $\beta_j := (\beta_{j,1}, \ldots, \beta_{j,n}) \in \zz^n$ such that $x_j = \prod_{i=1}^n y_i^{\beta_{j,i}}$, $j = 1, \ldots, n$. Then $\bar Y$ is the closure of the image of the map $\psi: \nktoruss{n-r} \to \kk^n$ given by
\begin{align*}
\psi (y_{r+1}, \ldots, y_n) :=  ( c'_1 \prod_{i=r+1}^n y_i^{\beta_{1,i}}, \ldots, 	 c'_n \prod_{i=r+1}^n y_i^{\beta_{n,i}}),
\end{align*}
where $c'_j := \prod_{i=1}^r c_i^{\beta_{j,i}}$, $j = 1, \ldots, n$. Let $\bar Y'$ be the closure of the image of the map $\psi' :\nktoruss{n-r} \to \kk^n$ given by
\begin{align*}
\psi'(y_{r+1}, \ldots, y_n) :=  (\prod_{i=r+1}^n y_i^{\beta_{1,i}}, \ldots, \prod_{i=r+1}^n y_i^{\beta_{n,i}})
\end{align*}
Then $\bar Y'$ is isomorphic to $\bar Y$ via the map $\rho: (x_1, \ldots, x_n) \mapsto (c'_1x_1, \ldots, c'_nx_n)$.
Let $\beta'_j := (\beta_{j,r+1}, \ldots, \beta_{j,n})$, $j = 1, \ldots, n$, and $\scrB' := \{\beta'_0, \ldots, \beta'_n\}$, where $\beta'_0$ is the origin in $\zz^{n-r}$. Let $\xbprime$ be the corresponding toric variety. In the notation of \cref{xa-thm} $\bar Y'$ is isomorphic to the affine open subset $\xbprime \cap U_{\beta'_0}$ of $\xbprime$. Since $\dim(\bar Y') = n-r$, it follows that the convex hull $\scrP'$ of $\scrB'$ in $\rr^{n-r}$ has dimension $n-r$. Let $\nu'_j := \langle \nu, \alpha_j \rangle$, $j = r+1, \ldots, n$, and $\nu' \in \rnnstar{n-r}$ be the element with coordinates $(\nu'_{r+1}, \ldots, \nu'_n)$. Then for each $j = 1, \ldots, n$, $\langle \nu', \beta'_j \rangle = \sum_{i = r+1}^n \beta_{j,i} \langle \nu, \alpha_i \rangle = \langle \nu, \sum_{i=1}^n \beta_{j,i} \alpha_i \rangle = \nu_j \geq 0$. It follows that $\min_{\scrP'}(\nu') = 0$ (since $\langle \nu', \beta'_0 \rangle = 0$). Since $\nu'_{r+1} > 0$, there is a facet $\scrQ'$ of $\scrP'$ containing $\In_{\nu'}(\scrP')$ such that the first coordinate of the inner normal with respect to the dual basis is positive (this follows e.g.\ from \cref{prop:edge-hull,normal-faces,stritctly-convex-fan}). Let $\scrC' := \scrB' \cap \scrQ'$, and $\corbit{\scrC'}$ be the corresponding torus invariant codimension one subvariety of $\xbprime$ (see \cref{xa-thm}). Let $Z' := \bar Y' \cap \corbit{\scrC'}$, and $Z := \rho(Z') \subseteq \bar Y$.

\begin{proclaim} \label{pi_I(a)-claim-2}
$\pi_I(a) \in Z \subseteq V(f_j: j \in J)$.
\end{proclaim}

\begin{proof}
Let $\eta \in \rnnstar{n-r}$ be the primitive inner normal to $\scrQ'$, and $(\eta_{r+1}, \ldots, \eta_n)$ be the coordinates of $\eta$ in the dual basis of $\rnnstar{n-r}$. For each $b = (b_{r+1}, \ldots, b_n) \in \nktoruss{n-r}$, consider the rational curve on $\bar Y'$ parametrized by
$$t \mapsto c'_b(t) := \psi'(b_{r+1}t^{\eta_{r+1}}, \ldots, b_nt^{\eta_n}) = (b^{\beta'_1}t^{\eta'_1}, \ldots, b^{\beta'_n}t^{\eta'_n})$$
where $\eta'_j := \langle \eta, \beta'_j \rangle$ for each $j = 1, \ldots, n$. Since $0 = \beta_0 \in \In_{\nu'}(\scrP') \subset \scrQ'$, it follows that $\min_{\scrP'}(\eta) = 0$, so that each $\eta'_j$ is nonnegative, and it is zero if and only if $\beta'_j \in \scrQ'$. In particular, $c'_b(0)$ is well defined, and is an element in $\kk^n$. Moreover, \cref{xa-thm} implies that the set $\{c'_b(0): b \in \nktoruss{n-r}\}$ is precisely $\bar Y' \cap \orbit{\scrC'}$; in particular it is a dense Zariski open subset of $Z'$. Therefore, in order to prove $Z \subseteq V(f_j: j \in J)$, it suffices to show that $f_j (\rho( c'_b(0))) = 0$ for each $j \in J$. Fix $j \in J$. Write $f_j$ as $f_j = f_{j,0} + f_{j,1} + \cdots $, where $f_{j,k}$ are $\nu$-homogeneous polynomials with $\nu(f_{j,0}) < \nu(f_{j,1}) < \cdots$. We will show that
\begin{align}
f_{j,k} (\rho(c'_b(0)))= 0 \label{f-j-k-0}
\end{align}
for each $k$. Let $m_j := \min_\nu(\scrA_j)$. Note that $\nu(f_{j,k}) \geq m_j$ for each $k$. At first consider the case that $\nu(f_{j,k}) = m_j$. This implies that $k = 0$, $\In_{\nu}(\scrA_j) \cap \supp(f_j)$ is nonempty, and $f_{j,k} = \In_{\scrA_j, \nu}(f_j)$. Since $\In_{\scrA_j, \nu}(f_j)(a) = 0$, in this case \cref{pi_I(a)-claim} implies that $f_{j,k}$ is identically zero on $\rho(c'_b(t))$, $t \in \kk$; in particular, \eqref{f-j-k-0} holds. Now fix $k$ such that $\nu(f_{j,k}) > m_j$. It suffices to show that
\begin{align}
\ord_t(f_{j,k} \circ \rho \circ  c'_b(t)) > 0 \label{ord-t-positive}
\end{align}
Pick $\alpha_0 \in \In_\nu(\scrA_j)$ and $\alpha \in \supp(f_{j,k})$. Then $\alpha - \alpha_0 \in \Pi$, and $\langle \nu, \alpha - \alpha_0 \rangle = \nu(f_{j,k}) - m_j > 0$. It follows that $\alpha - \alpha_0 = \sum_{j=1}^{r+1} m_j \alpha_j$ with integers $m_j$ such that $m_{r+1} > 0$. Since $x^{\alpha_j}|_Y \equiv c_j$ for $j = 1, \ldots, r$, and $x^{\alpha_{r+1}}|_Y \equiv  y_{r+1}|_Y$, it follows that
\begin{align*}
\ord_t(x^\alpha \circ \rho \circ c'_b(t))
	&= \ord_t(x^{\alpha_0} \circ \rho \circ c'_b(t)) + m_{r+1}\eta_{r+1}
\end{align*}

Since $\eta_{r+1} > 0$ (due to our choice of $\scrQ'$) and $\alpha_0 \in \znzero$, \eqref{ord-t-positive} follows. It remains to prove that $\pi_I(a) \in Z$. Consider the rational curve $C'$ on $\bar Y'$ parametrized by
\begin{align*}
t &\mapsto \psi'(c_{r+1}t^{\nu'_{r+1}}, \ldots, c_n t^{\nu'_n})
	= ((\prod_{i=r+1}^n c_i^{\beta_{1,i}}) t^{\nu_1}, \ldots, (\prod_{i=r+1}^n c_i^{\beta_{n,i}}) t^{\nu_n})

\end{align*}
Since $\In_{\nu'}(\scrP')$ is a face of $\scrQ'$, \cref{curve-to-orbit} implies that the center of the branch of $C'$ at $t = 0$ is contained in $Z'$. Note that $\rho(C')$ is precisely the curve $C$ from \cref{pi_I(a)-claim}. It follows from \cref{pi_I(a)-claim} that $\pi_I(a)$ is the center of the branch of $C$ at $t = 0$, and therefore it is on $Z$, as required.
\end{proof}
Since $\dim(Z) = n- r - 1 > n-|J|$, the lemma follows from \cref{pi_I(a)-claim-2,thm:pure-dimension}.
\end{proof}

\begin{cor}  \label{trivial-cor}
Let $I \in \dUA \setminus \{\emptyset\}$ and $\nu$ be a weighted order on $\kk[x_i: i \in I]$ such that $\nu$ is centered at $\Kstariprime$ for some $I' \subseteq I$, and $\In_{\Ai_1, \nu}(f_1|_{\Ki}), \ldots, \In_{\Ai_n,\nu}(f_n|_{\Ki})$ have a common zero $a \in \nktorus$. Then $\pi_{I'}(a) \in \Kstariprime$ is a non-isolated point of the zero-set $V(f_1, \ldots, f_n)$ of $f_1, \ldots, f_n$ on $\kk^n$.
\end{cor}

\begin{proof}
It is straightforward to check that $\pi_{I'}(a) \in V(f_1, \ldots, f_n)$. There is $J \supseteq I$ such that $\scrA$ is $\rj$-dependent, $|\tjA| = |J|$, and $|\titildeA| > |\tilde I|$ for each $\tilde I$ such that $I \subseteq \tilde I \subsetneq J$. Replacing the $f_j$ by $f_j|_{\Kii{J}}$ and applying \cref{reduction-lemma} reduces the corollary to the case that $I = [n]$. Then it follows from \cref{trivial-lemma}.

\end{proof}

\subsection{Proof of \cref{non-deg-1}} \label{non-deg-1-proof-section}
We divide the proof of \cref{non-deg-1} in three parts:
\subsubsection{Proof of the implication \eqref{non-deg-1-1} $\im$ \eqref{non-deg-1-0} from \cref{non-deg-1}}
We proceed as in the proof of \cref{bkk-sufficiency}. Pick $(f_1, \ldots, f_n), (g_1, \ldots, g_n) \in \scrL(\scrA)$ such that $\multgiso{U}> \multfiso{U}$. We will show that $f_1, \ldots, f_n$ are $(U,\scrA)$-degenerate. Define $h_j := (1-t)f_j + tg_j$, where $t$ is a new indeterminate. The set of isolated zeroes of $g_1, \ldots, g_n$ on $U$ is nonempty. \Cref{mult-deformation-global} implies that we can find an irreducible curve $C$ contained in the set of zeroes of $h_1, \ldots, h_n$ on $U \times \kk$ such that
\begin{prooflist}
\item \label{intersection-1} $C$ intersects $Z \times \{1\}$,
\item \label{generically-isolated} for generic $\epsilon \in \kk$, the set $C_\epsilon := C \cap (U \times \{\epsilon\})$ is nonempty, and each point of $C_\epsilon$ is an isolated zero of $h_1|_{t = \epsilon}, \ldots, h_n|_{t= \epsilon}$ on $U$; and
\item \label{property-at-zero}
\begin{prooflist}
\item \label{non-isolated-case} either there is $(z,0) \in C$ such that $z$ is a non-isolated zero of $f_1, \ldots, f_n$ on $U$,
\item \label{infinite-case} or $C$ has a ``point at infinity with respect to $U$'' at $t = 0$, i.e.\ if $\bar C$ is the closure of $C$ in $\pp^n \times \kk$, then there is $(z,0) \in \bar C$ such that $z \not\in U$.
\end{prooflist}
\end{prooflist}

\begin{claim} \label{tilde-I-claim}
Let $B$ be a branch of $\bar C \subset \pp^n \times \kk$. Let $\tilde I := I_B \cap [n]$ (where $I_B \subseteq [n+1]$ is defined as in \cref{IB-defn}) and $\tilde \nu$ be the restriction of $\nu_B$ to $\kk[x_i: i \in \tilde I]$.
\begin{enumerate}
\item \label{tilde-I-generic} $C_\epsilon \subseteq \Kstaritilde \times \{\epsilon\}$ for generic $\epsilon \in \kk$.
\item \label{tilde-I-UA} $\tilde I \in \iUA$.

\item \label{tilde-I-t0} Assume the center of $B$ is $(z,0)$ where $z \in \pp^n$. Then
\begin{enumerate}
\item \label{tilde-I-t0-zero} $\In_{\Atildei_j, \tilde \nu}(f_j|_{\Kitilde})$, $j = 1, \ldots, n$, have a common zero on $\nktorus$.
\item \label{tilde-I-t0-center-infinity} If $z \in \pp^n\setminus \kk^n$, then $\tilde I \neq \emptyset$ and $\tilde \nu$ is centered at infinity. In particular, $f_1, \ldots, f_n$ violate condition \ref{iUA-property} of $(U,\scrA)$-non-degeneracy with $I = \tilde I$ and $\nu = \tilde \nu$.
\item \label{tilde-I-t0-center-finity} Otherwise let $J$ be the (unique) subset of $[n]$ such that $z \in \Kstarii{J}$. Then either $\tilde I = J = \emptyset$, or $\tilde I \neq \emptyset$ and $\tilde \nu$ is centered at $\Kstarii{J}$. In particular, if $J \in \excU \cup \excA \cup \dUA$, then $f_1, \ldots, f_n$ violate property \ref{iUA-property} of $(U,\scrA)$-non-degeneracy with $I = \tilde I$, $I' = J$ and $\nu = \tilde \nu$.

\end{enumerate}

\end{enumerate}
\end{claim}

\begin{proof}
Assertion \eqref{tilde-I-generic} follows from the definition of $I_B$. Since $(h_1|_{t = \epsilon}, \ldots, h_n|_{t= \epsilon}) \in \scrL(\scrA)$ for all $\epsilon$, assertion \eqref{tilde-I-generic}, property \ref{generically-isolated} and \cref{non-isolated-lemma} imply that $\tilde I \not\in \excU \cup \excA$, and then \cref{hereditary-prop} implies that $\tilde I \in \iUA$. This proves assertion \eqref{tilde-I-UA}.
Now assume we are in the situation of assertion \eqref{tilde-I-t0}. Fix $j$, $1 \leq j \leq n$. Since $\nu_B(t|_{\Kib}) > 0$, it follows that
\begin{align*}
\In_{\Atildei_j, \tilde \nu}(f_j|_{\Kitilde})
	&=
	\begin{cases}
	\In_{\tilde \nu}(f_j|_{\Kitilde}) = \In_{\nu_B}(h_j|_{\Kib})& \text{if}\ \supp(f_j) \cap \In_{\tilde \nu}(\Atildei_j) \neq \emptyset,\\
	0 & \text{otherwise.}
	\end{cases}
\end{align*}
Assertion \eqref{tilde-I-t0-zero} now follows from \cref{branch-lemma-IB}. If $z \in \pp^n \setminus \kk^n$, then there is at least one $j$ such that $1/x_j$ is a regular function near $z$ which vanishes at $z$. This $j$ has to be in $\tilde I$ and $\tilde \nu(x_j)$ has to be negative, which proves the first statement of part \eqref{tilde-I-t0-center-infinity}. The second statement then follows from assertions \eqref{tilde-I-UA} and \eqref{tilde-I-t0-zero}. The first statement of part \eqref{tilde-I-t0-center-finity} is obvious. If $J \in \excU \cup \excA \cup \dUA$, then assertion \eqref{tilde-I-UA} and the first statement of part \eqref{tilde-I-t0-center-finity} imply that $\tilde I \neq \emptyset$, and then the second statement follows from assertions \eqref{tilde-I-UA} and \eqref{tilde-I-t0-zero}.

\end{proof}

Now we resume the proof of the implication \eqref{non-deg-1-1} $\im$ \eqref{non-deg-1-0} from \cref{non-deg-1}. At first assume \ref{non-isolated-case} holds. Pick a branch $B$ of $C$ centered at $(z,0)$ and let $\tilde I, J, \tilde \nu$ be as in part \eqref{tilde-I-t0-center-finity} of \cref{tilde-I-claim}. Part \eqref{tilde-I-t0-center-finity} of \cref{tilde-I-claim} implies that $f_1, \ldots, f_n$ are $(U, \scrA)$-degenerate if $J \in \excU \cup \excA \cup \dUA$. So assume $J \in \iUA$. Pick an irreducible curve $C'$ on $U$ such that $z \in C' \subseteq V(f_1, \ldots, f_n)$. Let $J'$ be the smallest subset of $[n]$ such that $C' \subseteq \Kii{J'}$. Since $J' \supseteq J$ and $J \not\in \excU \cup \excA$, it follows that $J' \not\in \excU \cup \excA$ as well. If $J' \in \iUA$, then \cref{branch-lemma-IB} implies that $f_1, \ldots, f_n$ violate condition \ref{iUA-property} of $(U,\scrA)$-non-degeneracy with $I = J'$ and some weighted order $\nu$ on $\kk[x_j: j \in J']$ centered at infinity (take a branch $B'$ of $C'$ centered at infinity, and set $\nu = \nu_{B'}$). On the other hand, if $J' \in \dUA$, then $f_1, \ldots, f_n$ violate condition \ref{dUA-property} of $(U,\scrA)$-non-degeneracy with $I = J'$ and $I' = J$ (take a branch $B'$ of $C'$ centered at $z$ and take $\nu := \nu_{B'}$). This completes the proof of \eqref{non-deg-1-1} $\im$ \eqref{non-deg-1-0} in the case that \ref{non-isolated-case} holds. Now assume we are in case \ref{infinite-case}. Pick $z \in \pp^n\setminus U$ such that $(z,0) \in \bar C$. If $z \in \pp^n \setminus \kk^n$, then part \eqref{tilde-I-t0-center-infinity} of \cref{tilde-I-claim} implies that $f_1, \ldots, f_n$ are $(U,\scrA)$-degenerate. So assume $z \in \kk^n \setminus U$. Define $J$ as in part \eqref{tilde-I-t0-center-finity} of \cref{tilde-I-claim}. If $J \in \excU \cup \excA \cup \dUA$, then part \eqref{tilde-I-t0-center-finity} of \cref{tilde-I-claim} implies that $f_1, \ldots, f_n$ are $(U, \scrA)$-degenerate. So assume $J \in \iUA$. But then $f_1, \ldots, f_n$ violate condition \ref{U-property} of $(U,\scrA)$-non-degeneracy. This completes the proof of the implication \eqref{non-deg-1-1} $\im$ \eqref{non-deg-1-0}. \\

\subsubsection{Proof of the implication \eqref{non-deg-1-0} $\im$ \eqref{non-deg-1-1} from \cref{non-deg-1}}
Assume $\multAiso{U} > 0$ and pick an $(U,\scrA)$-degenerate system $f_1, \ldots, f_n \in \scrL(\scrA)$. We will show that $\multAiso{U} > \multfiso{U}$. Recall the definition of $\iUAstar$ from \eqref{iUAstar}.

\begin{claim} \label{case-degenerate}
There is $I \in \iUAstar$ such that one of the following holds:
\begin{enumerate}
\item \label{case-nonempty} Either $I$ is nonempty, and there is a weighted order $\nu$ on $\kk[x_i: i \in I]$ such that $\In_{\Ai_j,\nu}(f_j|_{\Ki})$, $j = 1, \ldots, n$, have a common zero $a \in \nktorus$, and one of the following holds:
\begin{enumerate}
\item \label{case-infinity} $\nu$ is centered at infinity,
\item \label{case-U} or $\nu$ is centered at $\Kstariprime$ for some $I' \subseteq I$, and $\pi_{I'}(a) \not\in U$,
\item \label{case-non-isolated} or $\nu$ is centered at $\Kstariprime$ for some $I' \subseteq I$ and $\pi_{I'}(a)$ is a non-isolated zero of $f_1, \ldots, f_n$.
\end{enumerate}
\item \label{case-isolated-but-not} Or there is an isolated point $a$ of $V(f_1, \ldots, f_n) \cap \Ki \cap U$ which is {\em not} isolated in $V(f_1, \ldots, f_n) \subset \kk^n$.
\end{enumerate}
\end{claim}

\begin{proof}
If $f_1, \ldots, f_n$ violate property \ref{U-property} of $(U, \scrA)$-non-degeneracy, then there is a common zero $a'$ of $f_1|_{\Kiprime}, \ldots, f_n|_{\Kiprime}$ on $\Kstariprime \setminus U$ for some $I' \in \iUA$. \Cref{reduction-lemma} then implies that the claim holds with case \eqref{case-U} holds. If property \ref{iUA-property} of $(U, \scrA)$-non-degeneracy fails with $I \in \iUAstar$, then either the claim holds with case \eqref{case-infinity}, or there is a weighted order $\nu$ on $\kk[x_i: i \in I]$ such that $\In_{\Ai_j, \nu}(f_j|_{\Ki})$ have a common zero $a \in \nktorus$, and $\nu$ is centered at $\Kstariprime$ for some $I' \in \excU \cup \excA \cup \dUA$. It is straightforward to check that $\pi_{I'}(a)$ is in the set $V$ of common zeroes of $f_1, \ldots, f_n$ on $\kk^n$. If $I' \in \excU$, then we are in case \eqref{case-U}, since $\pi_{I'}(a) \in \Kstariprime$ and $\Kstariprime \cap U = \emptyset$. If $I' \in \excA \cup \dUA$, then $\pi_{I'}(a)$ has to be a non-isolated point of $V$ due to \cref{non-isolated-lemma,hereditary-prop}, which is case \eqref{case-non-isolated}. Due to \cref{non-deg-2} the only case left to consider is that of $f_1, \ldots, f_n$ violating property \ref{dUA-property} of $(U,\scrA)$-non-degeneracy. Then there is $J \in \dUA$ and a weighted order $\eta$ on $\kk[x_j: j \in J]$ centered at $\Kstarii{J'}$ for some $J' \in \iUA$ such that $\In_{\Aj_1, \eta}(f_1|_{\Kii{J}}), \ldots, \In_{\Aj_n, \eta}(f_n|_{\Kii{J}})$ have a common zero $b$ on $\nktorus$. \Cref{trivial-cor} implies that $\pi_{J'}(b)$ is a non-isolated point of $V(f_1, \ldots, f_n)$. Pick the smallest subset $I$ of $[n]$ containing $J'$ such that $|\tiA| = |I|$. Since $J' \not\in \excU \cup \excA$, it follows that $I \not\in \excU \cup \excA$, and since $\scrA$ is not hereditarily $\rii{J'}$-dependent, it follows that $\scrA$ is $\ri$-independent; in particular, $I \in \iUAstar$. If $\pi_{J'}(b)$ is an isolated point of $V(f_1, \ldots, f_n) \cap \Ki$ (which is e.g.\ the case if $I = J' = \emptyset$), then case \eqref{case-isolated-but-not} holds with $a = \pi_{J'}(b)$. Otherwise picking a branch at $\pi_{J'}(b)$ of a curve contained in $V(f_1, \ldots, f_n) \cap \Ki$ and applying \cref{branch-lemma-IB,reduction-lemma} shows that case \eqref{case-non-isolated} holds.
\end{proof}

\begin{claim} \label{case-degenerate-2}
Assume case \eqref{case-isolated-but-not} of \cref{case-degenerate} holds. Then there is $(g_1, \ldots, g_n) \in \scrL(\scrA)$ such that $g_j|_{\Ki} = f_j|_{\Ki}$ for each $j$, and $a$ is an isolated point of $V(g_1, \ldots, g_n) \subset \kk^n$.
\end{claim}

\begin{proof}
Fix $\tilde I \supsetneq I$. Since $|\tiA| = |I|$ and $I \not\in \excA$, it follows that $\titildeA \setminus \tiA$ contains at least $|\tilde I| - |I|$ elements. For each weighted order $\nu$ on $\kk[x_i: i \in \tilde I]$ centered at $\Kstari$, choosing generically coefficients of $f_j$, $j \in \titildeA \setminus \tiA$, it can be ensured that $\In_{\Atildei_j, \nu}(f_j)$, $j \in \titildeA \setminus \tiA$, have no common zero $\tilde a$ on $\nktorus$ such that $\pi_I(\tilde a) = a$. The claim now follows due to \cref{branch-lemma-IB}.
\end{proof}

At first consider case \eqref{case-isolated-but-not} of \cref{case-degenerate}. Pick $(g_1, \ldots, g_n)$ as in \cref{case-degenerate-2}. Apply \cref{mult-deformation-global} to $X = U$ and $h_j = (1-t)f_j + tg_j$, $j = 1, \ldots, n$. It is straightforward to see that in this case $\{a\} \times \kk$ is an irreducible component of the curve $C$ from \cref{mult-deformation-global}, so that
\begin{align*}
\multfiso{U}  \leq \multfkniso < \multpiso{h_1|_{t=\epsilon}}{h_n|_{t=\epsilon}}{\kk^n}
\end{align*}
for generic $\epsilon \in \kk$. It follows that $\multfiso{U} < \multAkniso = \multAiso{U}$, as required. Now assume there are $I$, $\nu$ and $a$ as in case \eqref{case-nonempty} of \cref{case-degenerate}. We may assume \woutlog\ that $I = \tiA = \{1, \ldots, k\}$ for some $k$, $1 \leq k \leq n$.

\begin{claim} \label{case-degenerate-1}
There is $(g_1, \ldots, g_n) \in \scrL(\scrA)$ such that
\begin{enumerate}
\item \label{g-a-non-zero} $\In_{\Ai_j, \nu}(g_j|_{\Ki})(a) \neq 0$ for each $j = 1, \ldots, k$.
\item \label{g-b-zero} there is a common zero $b$ of $g_1, \ldots, g_k$ on $\Kstari \cap U$ such that
\begin{enumerate}
\item \label{g-b-isolated} $b$ is an isolated point of $V(g_1, \ldots, g_n) \subset \kk^n$, and
\item \label{f-b-non-zero} $f_j(b) \neq 0$ for each $j = 1, \ldots, k$.
\end{enumerate}
\end{enumerate}
\end{claim}

\begin{proof}
Let $\scrL'(\scrA)$ be the collection of all $(g_1, \ldots, g_n) \in \scrL(\scrA)$ such that
\begin{prooflist}
\item \label{g-bkk-non-deg} $g_1|_{\Kii{J}}, \ldots, g_n|_{\Kii{J}}$ are $\Aj$-non-degenerate (in the sense of \cref{defn:b-non-deg}) for all $J \supset I$, and
\item \label{g-U-non-zero} there is no common zero of $g_1, \ldots, g_k$ on $\Kstari \setminus U$.
\end{prooflist}
Since $I \not\in \excU \cup \excA$, \cref{bkk-0,intersection-lemma} imply that $\scrL'(\scrA)$ contains a nonempty Zariski open subset of $\scrL(\scrA)$. It follows that $\scrL'_a(\scrA) := \{(g_1, \ldots, g_n) \in \scrL'(\scrA): \In_{\Ai_j, \nu}(g_j|_{\Ki})(a) \neq 0$ for each $j = 1, \ldots, k\}$ also contains a nonempty Zariski open subset of $\scrL(\scrA)$. Since $\scrA$ is $\ri$-independent, it follows that the ($k$-dimensional) mixed volume of $\conv(\scrA_j) \cap \ri$, $j = 1, \ldots, k$, is nonzero. Due to \ref{g-U-non-zero}, the arguments of \cref{claim:dominant} then imply that we can find $(g_1, \ldots, g_n) \in \scrL'_a(\scrA)$ and a common zero $b$ of $g_1, \ldots, g_n$ on $\Kstari \cap U$ such that $f_j(b) \neq 0$ for each $j = 1, \ldots, k$. Property \ref{g-bkk-non-deg} together with \cref{branch-lemma-IB} then imply that $b$ must be an isolated zero of $g_1, \ldots, g_n$ on $U$.
\end{proof}

Now we follow the process from \cref{modified-proof}. Fix integers $\nu'_j > \nu_j := \nu(x_j)$. Let $C$ be the rational curve on $\kk^n$ parametrized by $c(t):= (c_1(t), \ldots, c_n(t)) :\kk \to \kk^n$ given by
\begin{align*}
c_j(t) :=
\begin{cases}
 a_jt^{\nu_j} + (b_j - a_j)t^{\nu'_j}
 	& \text{if}\ 1 \leq j \leq k,\\
 0
 	& \text{otherwise.}
\end{cases}
\end{align*}
Let $m_j := \min_{\Ai_j}(\nu)$, $1 \leq j \leq k$. Define
\begin{align*}
h_j &:=
\begin{cases}
 t^{-m_j}f_j(c(t))g_j - t^{-m_j}g_j(c(t))f_j
 	& \text{if}\ 1 \leq j \leq k,\\
 (1-t)f_j + tg_j
 	& \text{otherwise.}
\end{cases}
\end{align*}
Assertion \eqref{g-a-non-zero} of \cref{case-degenerate-1} implies that each $h_j|_{t=0}$ is a nonzero constant times $f_j$, and assertion \eqref{g-b-zero} of \cref{case-degenerate-1} implies that each $h_j|_{t=1}$ is a nonzero constant times $g_j$ and $c(1) = b$ is an isolated zero of $g_1, \ldots, g_n$ on $U$. The assumptions of case \eqref{case-nonempty} of \cref{case-degenerate} implies that the center of $C$ at $t = 0$ is either out of $U$, or it is a non-isolated zero of $f_1, \ldots, f_n$. Since each $h_j$ vanishes on the curve $C' := \{(c(t),t): t \in \kk\} \subset \kk^{n+1}$, assertion \eqref{non-max-condition} of \cref{mult-deformation-global} implies that $\multfiso{U} < \multntorusiso{h_1|_{t=\epsilon}}{h_n|_{t=\epsilon}}$ for generic $\epsilon \in \kk$. It follows that $\multfiso{U} < \multAiso{U}$, as required.\\

\subsubsection{Proof that $\scrN(U, \scrA)$ is a nonempty Zariski open subset of $\scrL(\scrA)$}
As in \cref{bkk-existential-section} we call $\scrB = (\scrB_1, \ldots, \scrB_n)$ a {\em face} of $\scrA$ and write $\scrB \preceq \scrA$, if there is $\nu \in \rnstar$ such that $\scrB_j = \In_\nu(\scrA_j)$ for each $j$; in that case we also write $\scrB := \In_\nu(\scrA)$ and we say that $\scrB$ is {\em centered at infinity} (respectively, {\em centered at $\Kstari$} for some $I \subseteq [n]$) if $\nu$ is centered at infinity (respectively, centered at $\Kstari$). Moreover, if $g$ is a polynomial supported at $\scrA_j$, we write $g_{\scrB_j}$ for $\In_{\scrA_j, \nu}(g)$. Consider the systems of polynomials that violate either property \ref{iUA-property} or property \ref{dUA-property} of $(U, \scrA)$-non-degeneracy: pick $I \subseteq [n]$ and a weighted order $\nu$ on $\kk[x_i: i \in I]$. Let $\scrB := \In_\nu(\Ai)$ and $\DBI$ be the set of all $(g_1, \ldots, g_n) \in \scrL(\scrA)$ such that there is a common root of $(g_j|_{\Ki})_{\scrB_j}$, $j = 1, \ldots, n$, on $\nktorus$. If $(f_1, \ldots, f_n) \in \scrL(\scrA)$ is in the closure of $\DBI$, then \cref{DJBbar-closed} implies that $(f_1, \ldots, f_n) \in \DBprimeI$ for some $\scrB' \preceq \scrB$. Note that
\begin{prooflist}
\item \label{B-at-infinity} If $\scrB$ is centered at infinity, then $\scrB'$ is also centered at infinity.
\item If $\scrB$ is centered at $\Kstariprime$ for some $I' \in  \excU \cup \excA$, then $\scrB'$ is also centered at $\Kstarii{I''}$ for some $I'' \in  \excU \cup \excA$.

\item \label{B-d-B'-I} If $\scrB$ is centered at $\Kstariprime$ for some $I' \subseteq [n]$, then for each $j = 1, \ldots, n$,
\begin{align*}
f_j|_{\Kiprime}
    &=
    \begin{cases}
    f_{j, \scrB_j}
        &\text{if}\ \scrB_j \cap \riprime \neq \emptyset,\\
    0 &\text{otherwise.}
    \end{cases}
\end{align*}
It follows that
\begin{prooflist}
\item if $I' \in \dUA$ and $\scrB'$ is centered at $\Kstarii{I''}$ for some $I'' \in \iUA$, then $(f_1, \ldots, f_n)$ violate property \ref{dUA-property} of $(U, \scrA)$-non-degeneracy with $I, I'$ replaced respectively by $I',I''$;
\item if $I' \in \iUA$ and $\scrB'$ is centered at infinity, then $(f_1, \ldots, f_n)$ violate property \ref{iUA-property} of $(U, \scrA)$-non-degeneracy with $I, I'$ replaced respectively by $I',I''$;
\item if $I' \in \iUA$ and $\scrB'$ is centered at $\Kstarii{I''}$ for some $I'' \in \dUA$, then $(f_1, \ldots, f_n)$ violate property \ref{iUA-property} of $(U, \scrA)$-non-degeneracy with $I, I'$ replaced respectively by $I',I''$;
\end{prooflist}
\end{prooflist}
It follows from these observations that the set of systems which violate at least one of the properties \ref{iUA-property} and \ref{dUA-property} of $(U, \scrA)$-non-degeneracy is Zariski closed in $\scrL(\scrA)$. Now we tackle property \ref{U-property}. Pick $I \in \iUA$ and let $\DUI$ be the set of all $(g_1, \ldots, g_n) \in \scrL(\scrA)$ such that there is a common root of $g_1, \ldots, g_n$ on $\Kstari \setminus U$. If $(f_1, \ldots, f_n) \in \scrL(\scrA)$ is in the closure of $\DUI$, then the arguments from the proof of \cref{DJBbar-closed} imply that there is a weighted order $\nu$ on $\kk[x_i: i \in I]$ and a common zero $a$ of $\In_{\Ai_j, \nu}(f_j|_{\Ki})$, $j = 1, \ldots, n$, on $\nktorus$, such that
\begin{prooflist}[resume]
\item \label{U-nu-infty} either $\nu$ is centered at infinity, in which case $f_1, \ldots, f_n$ violate property \ref{iUA-property} of $(U, \scrA)$-non-degeneracy,
\item \label{U-nu-I'} or $\nu$ is centered at $\Kstariprime$ for some $I' \subseteq I$, and $\pi_{I'}(a) \in \Kstariprime \setminus U$; in this case $f_1, \ldots, f_n$ violate property \ref{U-property} (with $I$ replaced by $I'$) of $(U, \scrA)$-non-degeneracy if $I' \in \iUA$, and property \ref{iUA-property} of $(U, \scrA)$-non-degeneracy if $I' \in \dUA$.
\end{prooflist}
It follows that the set of $(U, \scrA)$-degenerate systems is Zariski closed in $\scrL(\scrA)$, as required. \qed

\section{Weighted B\'ezout theorem: general version} \label{weighted-application}
\subsection{Weighted B\'ezout theorem II: all \bm{$\omega(f_j)\geq 0$}}
For weights $\omega = (\omega_1, \ldots, \omega_n)$ to be applicable in weighted B\'ezout theorem (\cref{wt-bezout-1}), each $\omega_i$ has to be positive. In this section we replace this condition by a weaker one - that $\omega(f_j)$ has to be nonnegative for each $j$. This opens up a new possibility: if some $\omega_i$ is nonpositive, then the set of polynomials $f$ with $\omega(f)$ bounded above by a given integer will be an infinite dimensional vector space over $\kk$, and the number of (isolated) solutions of $n$ such polynomials can be arbitrarily large. Therefore to estimate number of solutions one has to bound the degree in each $x_i$ such that $\omega_i \leq 0$. It is natural then to consider, given an integer $d$, and a nonnegative integer $m_i$ for each $i$ such that $\omega_i \leq 0$, the set of polynomials supported at the polytope
\begin{align*}
\scrP(\omega, d, \vec m)
	&:= \{
		\alpha \in \rr^n: \langle \omega, \alpha \rangle \leq d,\
		\alpha_i \geq 0,\ i = 1, \ldots, n,\
		\alpha_i \leq m_i,\ i \in I_0 \cup I_-
		\}
\end{align*}
where $I_- := \{i: \omega_i < 0\}$ and $I_0 := \{i : \omega_i = 0\}$ (see \cref{fig:pomegadm}). The reason for our restriction to the case of nonnegative $\omega(f_j)$ is the following observation:

\begin{figure}[h]
\def\colorone{red}
\def\colortwo{blue}
\def\colorg{green}

\def\shiftx{7.5}
\def\colorzero{green}
\def\colorone{red}
\def\colortwo{blue}
\def\colorthree{yellow}
\def\opazero{0.5}
\def\viewx{120}
\def\viewy{30}
\def\titlex{2}
\def\titley{-1}
\def\scalefactor{0.6}
\begin{center}
\begin{subfigure}[b]{0.3\textwidth}
\begin{tikzpicture}[scale=\scalefactor]

\pgfplotsset{every axis title/.append style={at={(0,-0.2)}}, view={\viewx}{\viewy}, axis lines=middle, enlargelimits={upper}}

\begin{axis}
	\addplot3 [draw, ultra thick, fill=\colortwo,opacity=\opazero] coordinates{(0,0,1) (0,1,0) (1,0,0) (0,0,1)};

\end{axis}
\end{tikzpicture}
\caption{$\omega = (1,1,1)$,\\ $d = 1$}  \label{fig:pomega-ppp}
\end{subfigure}
\begin{subfigure}[b]{0.3\textwidth}
\begin{tikzpicture}[scale=\scalefactor]
\pgfplotsset{every axis title/.append style={at={(0,-0.2)}}, view={\viewx}{\viewy}, axis lines=middle, enlargelimits={upper}}

\begin{axis}
\addplot3[ thick, draw, fill=\colortwo,opacity=\opazero] coordinates{(1,0,0) (2,1,0) (2,1,1) (1,0,1)};
\addplot3[ thick, draw, fill=\colorone,opacity=\opazero] coordinates{(0,1,0) (2,1,0) (2,1,1) (0,1,1)};

\addplot3[ thick, draw, fill=\colorg,opacity=\opazero] coordinates{(1,0,1) (0,0,1) (0,1,1) (2,1,1)};

\end{axis}
\end{tikzpicture}
\caption{$\omega = (1,-1,0)$,\\ $d = 1$, $m_2 = m_3 = 1$}  \label{fig:pomega-p0n}
\end{subfigure}
\begin{subfigure}[b]{0.3\textwidth}
\begin{tikzpicture}[scale=\scalefactor]
\pgfplotsset{every axis title/.append style={at={(0,-0.2)}}, view={\viewx}{\viewy}, axis lines=middle, enlargelimits={upper}}

\begin{axis}
\addplot3[ thick, draw, fill=\colortwo,opacity=\opazero] coordinates{(1,0,0) (2,0,1) (3,1,1) (2,1,0)};
\addplot3[ thick, draw, fill=\colorone,opacity=\opazero] coordinates{(3,1,1) (2,1,0) (0,1,0) (0,1,1)};
\addplot3[ thick, draw, fill=\colorg,opacity=\opazero] coordinates{(3,1,1) (2,0,1) (0,0,1) (0,1,1)};

\end{axis}
\end{tikzpicture}
\caption{$\omega = (1,-1,-1)$,\\ $d = 1$, $m_2 = m_3 = 1$}  \label{fig:pomega-pnn}
\end{subfigure}
\caption{$P(\omega,d,\vec m)$ for different $\omega,d,\vec m$}  \label{fig:pomegadm}
\end{center}
\end{figure}

\begin{prop} \label{sum-pomega}
Let $d_1, \ldots, d_k \in \zz$ and $m_{j,i} \in \zzero$, $j = 1, \ldots, k$, $i \in I_0 \cup I_-$. If $d_j \geq 0$ for each $j$, then
\begin{align}
\sum_j \scrP(\omega, d_j, \vec m_j)
	= \scrP(\omega, \sum_j d_j, \sum_j \vec m_j)
\label{sumpomega}
\end{align}
\end{prop}

\begin{proof}
$\scrP(\omega, d_j, \vec m_j)$ is the product of an $(n-|I_0|)$-dimensional polytope of the same form with the $I_0$-dimensional box formed by the product over all $i \in I_0$ of the closed intervals from $0$ to $m_{j,i}$ on $x_i$-axis. It is then straightforward to see that to prove \eqref{sumpomega} it suffices to prove it under the additional condition that $I_0 = \emptyset$. So assume $I_0 = \emptyset$. Let $m := \max_{i,j}m_{j,i}$ and define
\begin{align*}
x'_i	&:=
	\begin{cases}
		x_i & \text{if}\ i \not\in I_- \\
		m-x_i & \text{if}\ i \in I_-
	\end{cases}\\
\omega'_i &:= |\omega_i|,\ i = 1, \ldots, n.
\end{align*}

It is straightforward to check that in $(x'_1, \ldots, x'_n)$-coordinates, in the notation of \cref{exercise:omegadm}, up to a reordering of the $x_i$ if necessary,
\begin{align}
\scrP(\omega, d_j, \vec m_j)
	&= \scrQ(\vec \omega', d'_j, \vec m_j) + (m'_{j,1}, \ldots, m'_{j,n})
\label{pomega=qomega'}
\end{align}
where
\begin{align*}
d'_j
	&:= d_j + \sum_{i \in I_-}|\omega_i|m_i,\ \text{and}\\
m'_{j,i}
	&:=
	\begin{cases}
		0 & \text{if}\ i \not\in I_- \\
		m - m_{j,i} & \text{if}\ i \in I_-
	\end{cases}
\end{align*}
It then follows from \cref{exercise:omegadm} that
\begin{align*}
\sum_j \scrP(\omega, d_j, \vec m_j)
	= \scrQ(\vec \omega', \sum_j d'_j, \sum_j \vec m_j) + \sum_j (m'_{j,1}, \ldots, m'_{j,n})
	=  \scrP(\omega, \sum_j d_j, \sum_j \vec m_j)
\end{align*}
as required.
\end{proof}

\begin{rem}
\eqref{sumpomega} may fail to hold if $d_j < 0$ for some $j$. Indeed, it follows from \cref{exercise:omegadm} and \eqref{pomega=qomega'} that with $\omega = (-1, -1, 1)$, $m_{1,1} = m_{1,2} = 1$ and $m_{2,1} = m_{2,2} = 3$, $\scrP(\omega, 1, \vec m_1) + \scrP(\omega, -3, \vec m_1) \subsetneq \scrP(\omega, -2, \vec m_1 + \vec m_2)$.
\end{rem}
We will now compute the mixed volume of $\scrP(\omega, d_j, \vec m_j)$, $j = 1, \ldots, n$. \Cref{exercise:vol-omegadm} and \eqref{pomega=qomega'} imply that
\begin{align}
\vol(\scrP(\omega, d, \vec m))
	&=  \frac{\prod_{i \in I_0}m_i}{(n-|I_0|)!\prod_{i \not\in I_0} |\omega_i|}
	   \sum_{\substack{I \subseteq I_- \\ d + \sum_{i \in I}|\omega_i| m_i > 0}} (-1)^{|I_-| - |I|}
	   (d + \sum_{i \in I}|\omega_i| m_i)^{n - |I_0|}
\label{volume-pomega}
\end{align}
In particular, if $d \geq 0$, then
\begin{align*}
\vol(\scrP(\omega, d, \vec m))
	&=  \frac{\prod_{i \in I_0}m_i}{(n-|I_0|)!\prod_{i \not\in I_0} |\omega_i|}
	   \sum_{I \subseteq I_- } (-1)^{|I_-| - |I|}
	   (d + \sum_{i \in I}|\omega_i| m_i)^{n - |I_0|}
\end{align*}
Therefore \cref{sum-pomega} implies that for each $d_1, \ldots, d_n, \lambda_1, \ldots, \lambda_n \geq 0$,
\begin{align*}
\vol(\sum_{j=1}^n \lambda_j\scrP(\omega, d_j, \vec m_j))

	&=  \frac{\prod_{i \in I_0}( \sum_{j=1}^n \lambda_j m_{j,i})}
		{(n-|I_0|)!\prod_{i \not\in I_0} |\omega_i|}
	   \sum_{I \subseteq I_- } (-1)^{|I_-| - |I|}
	   ( \sum_{j=1}^n \lambda_j (d_j + \sum_{i \in I}|\omega_i| m_{j,i}))^{n - |I_0|}
\end{align*}
The mixed volume of $\scrP(\omega, d_1, \vec m_1), \ldots, \scrP(\omega, d_n, \vec m_n)$ is the coefficient of $\lambda_1\cdots \lambda_n$ in the right hand side of the above expression. For each $J_0 \subseteq [n]$ such that $|J_0| = |I_0|$, the coefficient of $\prod_{j \in J_0} \lambda_j$ in $\prod_{i \in I_0}(\sum_{j=1}^n \lambda_j m_{j,i})$ is the {\em permanent} of the $|I_0| \times |I_0|$ matrix
\begin{align}
D_{I_0,J_0}
	&:=
	  \begin{pmatrix}
	  m_{j_1,i_1} & \cdots & m_{j_1, i_k}\\
	  \vdots & & \vdots \\
	  m_{j_k, i_1} & \cdots & m_{j_k,i_k}
	  \end{pmatrix}
\label{DI0J0}
\end{align}
where $k := |I_0| = |J_0|$ and $i_1, \ldots, i_k$ (respectively, $j_1, \ldots, j_k$) are elements of $I_0$ (respectively, $J_0$). (Note that $\perm(D_{I_0,J_0})$ does not depend on the ordering of the elements of $I_0$ or $J_0$. If $I_0 = J_0 = \emptyset$, then $D_{I_0,J_0}$ is the empty matrix, and its permanent is by convention $1$.) On the other hand, the coefficient of $\prod_{j \not\in J_0} \lambda_j$ in $(\sum_{j=1}^n \lambda_j (d_j + \sum_{i \in I}|\omega_i| m_{j,i}))^{n - k}$ is $(n-k)!\prod_{j \not\in J_0}(d_j + \sum_{i \in I}|\omega_i| m_{j,i})$. Combining all these together yields:

\begin{prop}\label{prop:mvpomega}
Let $d_1, \ldots, d_n \in \zz$ and $m_{j,i} \in \zzero$, $j = 1, \ldots, k$, $i \in I_0 \cup I_-$. If $d_j \geq 0$ for each $j$, then the mixed volume of $\scrP(\omega, d_1, \vec m_1), \ldots, \scrP(\omega, d_n, \vec m_n)$ is
\begin{align*}
\sum_{\substack{J_0 \subseteq [n]\\|J_0| = |I_0|}} \perm(D_{I_0,J_0})
	\sum_{I \subseteq I_-} (-1)^{|I_-| - |I|}
		\cfrac{
			\prod_{j \in [n] \setminus J_0} (d_j + \sum_{i \in I_-}|\omega_i|m_{j,i})
		}{
			\prod_{i \in [n] \setminus I_0}|\omega_i|}
\end{align*}
\end{prop}

The following result describes the faces of $\scrP(\omega, d, \vec m)$. Its proof is left as an exercise. Let $\eta \in \rnstar$ with coordinates $(\eta_1, \ldots, \eta_n)$ with respect to the basis dual to the standard basis of $\rr^n$.

\begin{prop} \label{pomega-face}
Define $M := \sup\{\eta_i/\omega_i: \omega_i > 0\}$. Assume $d \geq 0$. Then
\begin{enumerate}
\item If $M \leq 0$ (which is the case if in particular $\omega_i \leq 0$ for each $i$), then $\ld_\eta(\scrP(\omega, d, \vec m))$ is the set of all $(\alpha_1, \ldots, \alpha_n) \in \scrP$ such that
\begin{align*}
\alpha_i
	&=
	\begin{cases}
	0   &\text{if}\ \eta_i < 0,\\
    m_i &\text{if}\ \eta_i > 0.
    \end{cases}
\end{align*}
\item If $M > 0$, then $\ld_\eta(\scrP(\omega, d, \vec m))$ is the set of all $(\alpha_1, \ldots, \alpha_n) \in \scrP$ such that $\sum_i \alpha_i \omega_i = d$ and
\begin{align*}
\alpha_i
	&=
	\begin{cases}
	0   &\text{if}\ (\omega_i = 0,\ \eta_i < 0)\
            \text{or}\ (\omega_i > 0,\ \cfrac{\eta_i}{\omega_i} < M)\
            \text{or}\ (\omega_i < 0,\ \cfrac{\eta_i}{\omega_i} > M), \\
    m_i &\text{if}\ (\omega_i = 0,\ \eta_i > 0)\
            \text{or}\ (\omega_i < 0,\ \cfrac{\eta_i}{\omega_i} < M). \qed
    \end{cases}
\end{align*}
\end{enumerate}
\end{prop}

We are now ready to prove the second version of the weighted B\'ezout theorem. Let $f_j$ be polynomials supported at $\scrP(\omega, d_j, \vec m_j)$, $j = 1, \ldots, n$. For each $I \subseteq I_0 \cup I_-$, define
\begin{align*}
\scrL_{\vec m_j, I}
	&:=
	\{
	\alpha = (\alpha_1, \ldots, \alpha_n): \alpha_i = m_{j,i}\ \text{for each}\ i \in I
	\} \\
\scrL_{\omega, d_j, \vec m_j, I}
	&:= \{\alpha \in \scrL_{\vec m_j, I}: \sum_{i=1}^n \omega_i \alpha_i = d_j \}
\end{align*}
Denote the coefficient of $x^\alpha$ in $f_j$ by $c_{j\alpha}$, i.e.\ $f_j = \sum_{\alpha} c_{j\alpha}x^\alpha$. Write
\begin{align*}
\ld_{\vec m_j, I}(f_j)
	&:= \sum_{\alpha \in \scrL_{\vec m_j, I}} c_{j\alpha}x^\alpha,\\
\ld_{\omega, d_j, \vec m_j, I}(f_j)
	&:= \sum_{\alpha \in \scrL_{\omega, d_j, \vec m_j, I}} c_{j\alpha}x^\alpha
\end{align*}

\begin{thm}[Weighted B\'ezout theorem II] \label{wt-bezout-21}
Given polynomials $f_1, \ldots, f_n$ in $(x_1, \ldots, x_n)$ and any weighted degree $\omega$ on $\kk[x_1, \ldots, x_n]$, the number $N$ of isolated solutions of $f_1, \ldots, f_n$ on $\kk^n$ satisfies:
\begin{align}
N
	&\leq
		\sum_{\substack{J_0 \subseteq [n]\\|J_0| = |I_0|}} \perm(D_{I_0,J_0})
		\sum_{I \subseteq I_-} (-1)^{|I_-| - |I|}
		\cfrac{
			\prod_{j \in [n] \setminus J_0} \left(\max\{\omega(f_j),0\} + \sum_{i \in I_-}|\omega_i|\deg_{x_i}(f_j) \right)
		}{
			\prod_{i \in [n] \setminus I_0}|\omega_i|}
\label{wt-II}
\end{align}
where $D_{I_0,J_0}$ is defined as in \eqref{DI0J0} with $m_{j,i} := \deg_{x_i}(f_j)$. More generally, if $d_1, \ldots, d_n$ and $m_{j,i}$ are {\em nonnegative} integers such that each $d_j \geq \omega(f_j)$ and $m_{j,i} \geq \deg_{x_i}(f_j)$, then
\begin{align}
N
	&\leq
		\sum_{\substack{J_0 \subseteq [n]\\|J_0| = |I_0|}} \perm(D_{I_0,J_0})
		\sum_{I \subseteq I_-} (-1)^{|I_-| - |I|}
		\cfrac{
			\prod_{j \in [n] \setminus J_0} \left(d_j + \sum_{i \in I_-}|\omega_i|m_{j,i} \right)
		}{
			\prod_{i \in [n] \setminus I_0}|\omega_i|}
\label{wt-II-1}
\end{align}

If the right hand side of \eqref{wt-II-1} is nonzero, i.e.\ $\scrP(\omega, d_1, \vec m_1), \ldots, \scrP(\omega, d_n, \vec m_n)$ are independent (which is the case e.g.\ if each $d_j$ and each $m_{j,i}$ is positive), then \eqref{wt-II-1} holds with an equality if and only if both of the following are true:
\begin{enumerate}
\item \label{I-condition} For each {\em nonempty} subset $I$ of $I_0 \cup I_-$, $\ld_{\vec m_j, I}(f_j)$, $j = 1, \ldots, n$, have no common zero on $\kk^n \setminus \bigcup_{i \in I} V(x_i)$.
\item \label{omegaI-condition} For each (possibly empty) subset $I$ of $I_0 \cup I_-$, $\ld_{\omega, d_j, \vec m_j, I}(f_j)$, $j = 1, \ldots, n$, have no common zero on $\kk^n \setminus \left(\cup_{i \in I} V(x_i)) \bigcup (\cap_{i \in I_+}V(x_i)) \right)$ (where $I_+ := \{i: \omega_i > 0\}$).
\end{enumerate}
\end{thm}

\begin{proof}
Inequalities \eqref{wt-II} and \eqref{wt-II-1} follow directly from \cref{extended-bkk-bound-0,prop:mvpomega}. To find the non-degeneracy conditions we apply \cref{non-deg-1,non-deg-2} with $\scrA := (\scrP(\omega, d_1, \vec m_1) \cap \zz^n, \ldots, \scrP(\omega, d_n, \vec m_n) \cap \zz^n)$ and $U := \kk^n$. It is straightforward to check that $\excA = \excU = \emptyset$, and if $\scrP(\omega, d_j, \vec m_j)$, $j = 1, \ldots, n$, are independent then $\dUA = \emptyset$ and $\iUAstar = [n]$. Therefore \cref{non-deg-1,non-deg-2} imply that if $\scrP(\omega, d_j, \vec m_j)$, $j = 1, \ldots, n$, are independent (which due to \cref{positively-mixed,prop:mvpomega} is equivalent to the right hand side of \eqref{wt-II-1} being nonzero), then \eqref{wt-II-1} holds with an equality if and only if the following holds:
\begin{align}
\parbox{0.69\textwidth}{
for each weighted order $\nu$ centered at infinity on $\kk[x_1, \ldots, x_n]$, there is no common zero of $\In_{\scrA_1, \nu}(f_1), \ldots, \In_{\scrA_n,\nu}(f_n)$ on $\nktorus$.
}\label{wt-non-deg-20}
\end{align}
Pick a weighted order $\nu$ centered at infinity on $\kk[x_1, \ldots, x_n]$. Now apply \cref{pomega-face} to $\eta := -\nu$. If $M := \sup\{\eta_i/\omega_i: \omega_i > 0\} \leq 0$, then $I := \{i \in I_0 \cup I_-: \eta_i > 0\}$ is nonempty and \cref{pomega-face} implies that
\begin{align*}
\In_{\scrA_j, \nu}(f_j)
    &= \ld_{\vec m_j, I}(f_j)|_{\{x_{i'} = 0: i' \in I'\}}
\end{align*}
where $I' := \{i \in [n]: \eta_i < 0\}$. Therefore \eqref{wt-non-deg-20} is equivalent to the condition that $\ld_{\vec m_j, I}(f_j)$, $j = 1, \ldots, n$, have no common zero on $\left( \kk^n \setminus \bigcup_{i \in [n] \setminus I'} V(x_i)\right) \cap \bigcap_{i' \in I'} V(x_{i'})$. Since it is possible for $I'$ to be any subset of $[n]\setminus I$, taking into account all such possibilities leads to condition \eqref{I-condition}. Now consider the case that $M > 0$. In this case define $I := \{i \in I_0: \eta_i > 0\} \cup \{i \in I_-: \eta_i/\omega_i < M\}$ and $I' := \{i \in I_0: \eta_i < 0\} \cup \{i \in I_-: \eta_i/\omega_i > M\} \cup \{i \in I_+: \eta_i/\omega_i < M\}$. \Cref{pomega-face} implies that
\begin{align*}
\In_{\scrA_j, \nu}(f_j)
    &= \ld_{\omega, d_j, \vec m_j, I}(f_j)|_{\{x_{i'} = 0: i' \in I'\}}
\end{align*}
It then follows as in the preceding case that to satisfy \eqref{wt-non-deg-20} for all choices of $I'$ is equivalent to condition \eqref{omegaI-condition}, and this completes the proof.
\end{proof}

\subsection{Weighted B\'ezout theorem III: the general case}
In this section we consider weighted B\'ezout theorem without any restriction on the $\omega_i$ or $\omega(f_j)$. In this generality we do not know of any compact expressions for either the mixed volume or the faces of $\scrP(\omega, d_j, \vec m_j)$. Therefore our version of general weighted B\'ezout theorem below is little more than direct application of the extension of Bernstein's theorem to the affine space.

\begin{thm}[Weighted B\'ezout theorem III] \label{wt-bezout-3}
Let $d_j\in \zz$ and $m_{j,i} \in \zzero$ be such that $d_j \geq \omega(f_j)$ for each $j$ and each $m_{j,i} \geq \deg_{x_i}(f_j)$. Given $I \subseteq I_-$, write $\badj(I) := \{j: \sum_{i \in I_-\setminus I} m_{j,i}\omega_i > d_j\}$. Let $\goodI := \{I \subseteq I_-: |\badj(\tilde I)| \leq |\tilde I|$ for each $\tilde I \subseteq I\}$. Then the number $N$ of isolated solutions of $f_1, \ldots, f_n$ on $\kk^n$ is
\begin{align}
N \leq
	\sum_{I \in \goodI, |\badj(I)| = |I|}
	\mv_{n-k}(\scrP_{j'_1} \cap \riprime, \ldots, \scrP_{j'_{n-k}} \cap \riprime)
	\times \multzero{\pi_I(\scrP_{j_1})}{\pi_I(\scrP_{j_k})}
\label{general-wt-formula}
\end{align}
where
\begin{itemize}
\item $\scrI' := [n]\setminus I$,
\item $\scrP_j$ are short for $\scrP(\omega, d_j, \vec m_j)$,
\item $j_1, \ldots, j_k$ (respectively, $j'_1, \ldots, j'_{n-k}$) are elements of $\badj(I)$ (respectively, $\{1, \ldots, n\} \setminus \badj(I)$), and
\item $\mv_{n-k}(\cdot, \ldots, \cdot)$ is the $(n-k)$-dimensional mixed volume.
\end{itemize}
The bound in \eqref{general-wt-formula} holds with an equality if and only if $f_1, \ldots, f_n$ are $(\kk^n, \scrA)$-non-degenerate, where $\scrA := (\scrP_1 \cap \zz^n, \ldots, \scrP_n \cap \zz^n)$.
\end{thm}

\begin{proof}
Follows immediately from \cref{extended-bkk-bound-0,non-deg-1,non-deg-2}.
\end{proof}

\section{Weighted multi-homogeneous B\'ezout theorem: general version} \label{weighted-multapplication}
In this section we generalize the weighted multi-homogeneous version of B\'ezout's theorem (\cref{multi-bezout}) by replacing the weighted degrees with positive weights by weighted degrees from \cref{wt-bezout-21}. As in the setting of \cref{multi-bezout}, let $\mscrI := (I_1, \ldots, I_s)$ be an ordered partition of $[n] := \{1, \ldots, n\}$, and for each $j = 1, \ldots, s$, let $\omega_j$ be a weighted degree on $\kk[x_k: k \in I_j]$ with weights $\omega_{j,k}$ for $x_k$, $k \in I_j$. Let $I_{j,+}$ (respectively, $I_{j,-}$) be the set of all $k \in I_j$ such that $\omega_{j,k} > 0$ (respectively, $\omega_{j,k} < 0$). Given nonnegative integers $d_{i,j} \geq \omega_j(f_i)$ and $m_{i,j,k} \geq \deg_{x_k}(f_i)$ for each $k \in I_{j,0} \cup I_{j,-}$, we consider the polytope
\begin{align*}
\scrP_i
    &:= \prod_{j = 1}^s \scrP(\omega_j, d_{i,j}, \vec m_{i,j})
\end{align*}
Let $n_j := |I_j|$, $j = 1, \ldots, s$. The mixed volume of $\scrP_1, \ldots, \scrP_n$ is the coefficient of $\lambda_1 \cdots \lambda_n$ in the polynomial
\begin{align*}
\vol_n(\sum_{i=1}^n \lambda_i \scrP_i)
	&= \vol_n(\sum_{i=1}^n \lambda_i \prod_{j=1}^s \scrP(\omega_j, d_{i,j}, \vec m_{i,j}))
	= \vol_n(\prod_{j=1}^s (\sum_{i=1}^n \lambda_i \scrP(\omega_j, d_{i,j}, \vec m_{i,j}))) \\
    &= \vol_n(\prod_{j=1}^s \scrP(\omega_j, \sum_{i=1}^n \lambda_i d_{i,j}, \sum_{i=1}^n \lambda_i \vec m_{i,j}))
    = \prod_{j=1}^s \vol_{n_j}(\scrP(\omega_j, \sum_{i=1}^n \lambda_i d_{i,j}, \sum_{i=1}^n \lambda_i \vec m_{i,j}))
\end{align*}
where the third equality follows from \cref{sum-pomega}. After a refinement of $\mscrI$ if necessary, we may, and will, assume that $\omega_{j,k} \neq 0$ for each $j,k$, i.e.\ $I_j = I_{j,+} \cup I_{j,-}$ for each $j$. Then \eqref{volume-pomega} implies that
\begin{align*}
\vol_n(\sum_{i=1}^n \lambda_i \scrP_i)
    &= \prod_{j=1}^s \left(
            \frac{1}{n_j!\prod_{k \in I_j} |\omega_{j,k}|}
        	   \sum_{I \subseteq I_{j,-} } (-1)^{|I_{j,-}| - |I|}
        	   ( \sum_{i=1}^n \lambda_i (d_{i,j} + \sum_{k \in I}|\omega_{j,k}| m_{i,j,k}))^{n_j}
            \right)
\end{align*}
Let $I_- := \bigcup_j I_{j,-}$. For each $I \subseteq I_-$, write
\begin{align}
d_{I,i,j}
    &:= d_{i,j} + \sum_{k \in I \cap I_j}|\omega_{j,k}| m_{i,j,k}
\end{align}
Let $\Omega := \{\omega_1, \ldots, \omega_s\}$, and $D(\Omega, \vec d, \vec m, I)$ be the following $n \times n$ matrix:
\vspace{2mm}

\begin{align*}
\begin{matrix}
D(\Omega, \vec d, \vec m, I)
	&:=
	  \begin{pmatrix}
	  \bovermat{$n_1$ times}{ d_{I,1,1}  & \cdots & d_{I,1,1}}
	  	&
	  \overmat{$\mkern-3.5mu\cdots$}{\cdots & \cdots}
	  	&
	  \bovermat{$n_s$ times}{ d_{I,1,s}  & \cdots & d_{I,1,s}}
	   	\\[0.5em]
	   	\vdots & & \vdots & & & \vdots & & \vdots \\
	   	d_{I,n, 1}  & \cdots & d_{I,n,1} & \cdots & \cdots & d_{I,n,s}  & \cdots & d_{I,n,s}
	   \end{pmatrix}
\end{matrix}
\end{align*}
The preceding discussion together with \cref{extended-bkk-bound-0} or \cref{equal-multpn} imply that
\begin{align}
\multfkniso
    \leq \multPkniso
    = \mv(\scrP_1, \ldots, \scrP_n)
	= \sum_{I \subseteq I_-} (-1)^{|I_{-}| - |I|}
        \frac{ \perm(D(\Omega, \vec d, \vec m, I))}
		 {(\prod_j n_j!) (\prod_{j,k}\omega_{j,k})}
		 \label{wt-multi-homogeneous-bound-II}
\end{align}
It is straightforward to check that the bound \eqref{wt-II-1} from weighted B\'ezout theorem II corresponds to the special case of \eqref{wt-multi-homogeneous-bound-II} in which $n_j = 1$ for all but possibly one $j \in \{1, \ldots, s\}$. We will now see that the criterion for attainment of this bound is an amalgam of the non-degeneracy criteria of weighted multi-homogeneous B\'ezout theorem (\cref{multi-bezout}) and weighted B\'ezout theorem II (\cref{wt-bezout-21}). Given $I \subseteq I_-$, $J \subseteq [s]$, and $l \in [n]$, let $\scrL_{\Omega, \vec d, \vec m, I, J, i}$ be the set of all $\alpha = (\alpha_1, \ldots, \alpha_n) \in \znzero$ such that
\begin{itemize}
\item $\alpha_k = m_{i,j,k}$ for each $j \in [s]$ and $k \in I_{j,-} \cap I$, and
\item $\sum_{k \in I_j} \omega_{j,k}\alpha_k = d_{i,j}$ for each $j \in J$.
\end{itemize}
Given $g = \sum_\alpha c_\alpha x^\alpha \in \kk[x_1, \ldots, x_n]$ supported at $\scrP_i$, define
\begin{align*}
\ld_{\Omega, \vec d, \vec m, I, J, i}(g)
    &:= \sum_{\alpha \in \scrL_{\Omega, \vec d, \vec m, I, J, i}} c_\alpha x^\alpha
\end{align*}
We are now ready to prove version II of the weighted multi-homogeneous B\'ezout theorem. But at first we recall the assumptions:
\begin{defnlist}
\item $\omega_{j,k} \neq 0$ for each $j = 1, \ldots, s$, and $k \in I_j$.
\item $d_{i,j} \geq \max\{\omega_j(f_i), 0\}$ for each $i = 1, \ldots, n$, and $j = 1, \ldots, s$.
\item $m_{i,j,k} \geq \deg_{x_k}(f_i)$ for each $i = 1, \ldots, n$, and $j = 1, \ldots, s$, and $k \in I_{j,-}$.
\end{defnlist}

\begin{thm}[Weighted multi-homogeneous B\'ezout theorem II] \label{multi-bezout-II}
Under the above assumptions the number of isolated solutions of polynomials $f_1, \ldots, f_n$ on $\kk^n$ is bounded by \eqref{wt-multi-homogeneous-bound-II}. This bound is exact if and only if the following holds: for each pair $I,J$ such that $I \subseteq I_-$, $J \subseteq [s]$, and at least one of $I$ and $J$ is nonempty, there is no common zero of $\ld_{\Omega, \vec d, \vec m, I, J, 1}(f_1), \ldots, \ld_{\Omega, \vec d, \vec m, I, J, n}(f_n)$ on $\kk^n \setminus \left(\cup_{i \in I} V(x_i)) \bigcup (\cup_{j \in J}\cap_{k \in I_{j,+}}V(x_k)) \right)$.
\end{thm}

\begin{proof}
This follows from \cref{extended-bkk-bound-0,non-deg-1,non-deg-2} via arguments similar to those in the proof of \cref{wt-bezout-21}.
\end{proof}

\section{Open problems}
\subsection{Systems with isolated zeroes on a given coordinate subspace}
Given finite subsets $\scrA_1, \ldots, \scrA_n$ of $\znzero$ and a coordinate subspace $\Ki$ of $\kk^n$, it is straightforward to identify if for {\em generic} $f_j$ supported at $\scrA_j$, there is any common zero of $f_1, \ldots, f_n$ on $\Ki$, and in case there are such points, if they are isolated in $V(f_1, \ldots, f_n)$ or not. This is the content of the next result, which is straightforward to prove from \cref{positively-mixed,non-isolated-lemma}.

\begin{prop}[Zeroes of generic systems] \label{generic-characterization}
Let $\scrA := (\scrA_1, \ldots, \scrA_n)$. For all $f = (f_1, \ldots, f_n) \in \scrL(\scrA)$, write $V(f) := V(f_1, \ldots, f_n) \subset \kk^n$ and $\VstarIf := V(f_1, \ldots, f_n) \cap \Kstari$.
\begin{enumerate}
\item The following are equivalent:
\begin{enumerate}

\item $\VstarIf= \emptyset$ for generic $f \in \scrL(\scrA)$.
\item $\scrA$ is $\ri$-dependent.
\end{enumerate}
\item The following are equivalent:
\begin{enumerate}
\item $\VstarIf \neq \emptyset$ and all points of $\VstarIf$ are isolated in $V(f)$ for generic $f \in \scrL(\scrA)$.
\item $\scrA$ is $\ri$-independent and $I \not\in \excA$.

\end{enumerate}
\item The following are equivalent:
\begin{enumerate}
\item $\VstarIf \neq \emptyset$ and all points of $\VstarIf$ are non-isolated in $V(f)$ for generic $f \in \scrL(\scrA)$.
\item $\scrA$ is $\ri$-independent and $I \in \excA$. \qed

\end{enumerate}
\end{enumerate}
\end{prop}

\begin{problem}[Existence of systems with given support and isolated zeroes on given coordinate subspace] \label{isolated-problem}
Characterize those $I \subseteq [n]$ for which there are $f_1, \ldots, f_n$ such that $\supp(f_j) = \scrA_j$, $j = 1, \ldots, n$, $\VstarI(f_1, \ldots, f_n) \neq \emptyset$ and all points (or some points) of $\VstarI(f_1, \ldots, f_n)$ are isolated in $V(f_1, \ldots, f_n)$.
\end{problem}

If $I$ is as in \cref{isolated-problem}, then
\begin{itemize}
\item \cref{non-isolated-lemma} implies that $I \not\in \excA$;
\item \cref{bkk-non-degenerate-thm,positively-mixed} imply that
\begin{itemize}
\item either $I$ is $\ri$-independent and $|\tiA| = |I|$ (i.e.\ $I \in \iUAstar$),
\item or $|\tiA| > |I|$ (in particular, $\scrA$ is $\ri$-dependent);
\end{itemize}
\item \cref{hereditary-prop} implies that $\scrA$ is not hereditarily $\ri$-dependent.
\end{itemize}
In the context of these observations \cref{isolated-problem} boils down to the following problem.

\addtocounter{thm}{-1}
\let\oldthethm\thethm
\renewcommand{\thethm}{\oldthethm$'$}
\begin{problem} \label{isolated-problem'}
If $I \not\in \excA$ and $\scrA$ is not hereditarily $\ri$-dependent and $|\tiA| > |I|$, does there exist $f_1, \ldots, f_n$ such that $\supp(f_j) = \scrA_j$ for each $j$, $\VstarI(f_1, \ldots f_n) \neq \emptyset$ and all points (or some points) of $\VstarI(f_1, \ldots, f_n)$ are isolated in $V(f_1, \ldots, f_n)$? If not, then characterize those $I \subseteq [n]$ which satisfy the hypothesis of the preceding question but fail the conclusion.
\end{problem}
\let\thethm\oldthethm

\subsection{Non-isolated zeroes and non-degeneracy}
In contrast to the case of $\nktorus$, \cref{ex-aff-non-deg-1} shows that for $f = (f_1, \ldots, f_n) \in \scrL(\scrA)$, the existence of non-isolated solutions of $f_1, \ldots, f_n$ does not automatically mean that $f_1, \ldots, f_n$ are $(\kk^n, \scrA)$-non-degenerate. More precisely, part \ref{aff-non-deg-1-1} of \cref{ex-aff-non-deg-1} shows that if $I \in \dUUA{\kk^n}$, then it is possible for $f_1, \ldots, f_n$ to be $(\kk^n,\scrA)$-non-degenerate even if $\VstarIf$ has non-isolated points. On the other hand, condition \ref{iUA-property} of $(U, \scrA)$-non-degeneracy implies that if $I \in \iUUA{\kk^n}$ and $\VstarIf$ has non-isolated points, then $f_1, \ldots, f_n$ are $(\kk^n,\scrA)$-degenerate. The question is if it is sufficient.

\begin{problem} \label{non-isolated-problem}
If $I \in \dUUA{\kk^n}$, does there exist $f = (f_1, \ldots, f_n)$ such that $\supp(f_j) = \scrA_j$ for each $j$, and $(V^*)^I(f)$ is nonempty (and due to \cref{hereditary-prop} necessarily positive dimensional), but $f_1, \ldots, f_n$ are $(\kk^n,\scrA)$-non-degenerate? If not, then characterize those $I \in \dUUA{\kk^n}$ for which there is no such $f \in \scrL(\scrA)$.
\end{problem}

\subsection{Simple criteria for equality of Li and Wang's bound}
Since the upper bound of Li and Wang from \eqref{mv-estimate} is so simple, it would be interesting to find simple criteria under which it holds with equality. \Cref{li-wang-condition} gives a characterization of all such scenarios, so the question is if it can be made ``more explicit'' in any sense, or if there are simpler criteria (e.g.\ as in \cref{conveniently-li-wang} or assertion \eqref{leq-n-2} of \cref{li-wang-condition}) which are sufficient. One possible criterion was proposed in \cite{rojas-wang} and \cite{rojas-toric}, but that turns out to be incorrect. Indeed, both \cite[Theorem 1]{rojas-wang} and \cite[Affine Point Theorem II]{rojas-toric} imply the following: if $\excA = \emptyset$ and the intersection of each $\scrA_j$ with each of the $n$ coordinate hyperplanes is nonempty, then Li and Wang's bound holds with equality. This is indeed the case for $n \leq 2$, but as the following example shows, it is false in higher dimensions.

\begin{example}
Let $f_1 := ax +by + cx^2$, $f_2 := a'x + b'y +c'x^2$, $f_3 := pz^kx + q$, where $a,b,c,a',b',c',p,q$ are generic elements in $\kk$ and $k \geq 1$. It is straightforward to check directly that on $\kk^n$ there are precisely $k$ solutions for $f_1 = f_2 = f_3 = 0$ and $2k$ solutions for $f_1 = f_2 = f_3 = t$ for generic $t \neq 0$, so that Li and Wang's bound fails for $\scrA_j := \supp(f_j)$, $j = 1, 2, 3$. Note that both conditions \ref{m_nu-condition} and \ref{dimension-condition} from \cref{li-wang-section} hold with the weighted degree $\nu$ on $\kk[x,y,z]$ corresponding to weights $x \mapsto k$, $y \mapsto k$, $z \mapsto -1$.
\end{example}

\subsection{``Compact'' formulae for general weighted and weighted multi-homogeneous versions of B\'ezout's theorem}
There are scenarios not covered in weighted B\'ezout theorem II (\cref{wt-bezout-21}) for which very similar bound exists, e.g.\ in the case that $|I_-| = 1$ (and no restriction that the $d_j$ have to be nonnegative). This motivates the question: is it possible to find a formula that is more explicit than the one from weighted B\'ezout theorem III (\cref{wt-bezout-3}), and which is more general than version II? That would also lead to a more general version of weighted multi-homogeneous B\'ezout theorem II.