<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">%~MoulinÃ© par MaN_auto v.0.33.8 (9c3f6e85) 2025-05-02 15:01:59
\documentclass[CRMATH,Unicode,XML,biblatex]{cedram}

\TopicFR{Analyse et gÃ©omÃ©trie complexes, GÃ©omÃ©trie algÃ©brique}
\TopicEN{Complex analysis and geometry, Algebraic geometry}

\addbibresource{CRMATH_Fassarella_20230711.bib}

\usepackage{amssymb}
\usepackage{mathtools}

\DeclareMathOperator{\euler}{\chi_\mathrm{top}}
\DeclareMathOperator{\sing}{Sing}
\DeclareMathOperator{\grad}{grad}
\DeclareMathOperator{\mv}{MV}
\DeclareMathOperator{\vol}{vol}
\DeclareMathOperator{\divisor}{div}
\DeclareMathOperator{\ord}{ord}

\newcommand{\Lcal}{\mathcal{L}}
\newcommand{\Ocal}{\mathcal{O}}
\newcommand{\Hcal}{\mathcal{H}}
\newcommand{\Bcal}{\mathcal{B}}
\newcommand{\toricscheme}{\mathcal{T}}

\newcommand{\Z}{\mathbb{Z}}
\renewcommand{\P}{\mathbb{P}}
\newcommand{\A}{\mathbb{A}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\R}{\mathbb{R}}

\newcommand{\SM}{\mathrm{SM}}
\newcommand{\csm}{c_{\SM}}

\newcommand{\dv}[2]{\frac{\partial{#1}}{\partial{#2}}}

\newcommand{\hf}{\hat{f}}

\newcommand*{\dd}{\mathrm{d}}

%%%---------------------------------------------------------------------------

%%% ARROWS

%%% \to macro
\renewcommand*{\to}{\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}}
%% Remplacer \xrightarrow{#1} par \overset{#1}{\longrightarrow}. 

%%% \mapsto macro
\let\oldmapsto\mapsto
\renewcommand*{\mapsto}{\mathchoice{\longmapsto}{\oldmapsto}{\oldmapsto}{\oldmapsto}}

%%%---------------------------------------------------------------------------

%%% BAR &amp; OVERLINE

%%% Better widebar macro
\makeatletter
\let\save@mathaccent\mathaccent
\newcommand*\if@single[3]{%
  \setbox0\hbox{${\mathaccent"0362{#1}}^H$}%
  \setbox2\hbox{${\mathaccent"0362{\kern0pt#1}}^H$}%
  \ifdim\ht0=\ht2 #3\else #2\fi
  }
%The bar will be moved to the right by a half of \macc@kerna, which is computed by amsmath:
\newcommand*\rel@kern[1]{\kern#1\dimexpr\macc@kerna}
%If there's a superscript following the bar, then no negative kern may follow the bar;
%an additional {} makes sure that the superscript is high enough in this case:
\newcommand*\widebar[1]{\@ifnextchar^{{\wide@bar{#1}{0}}}{\wide@bar{#1}{1}}}
%Use a separate algorithm for single symbols:
\newcommand*\wide@bar[2]{\if@single{#1}{\wide@bar@{#1}{#2}{1}}{\wide@bar@{#1}{#2}{2}}}
\newcommand*\wide@bar@[3]{%
  \begingroup
  \def\mathaccent##1##2{%
%Enable nesting of accents:
    \let\mathaccent\save@mathaccent
%If there's more than a single symbol, use the first character instead (see below):
    \if#32 \let\macc@nucleus\first@char \fi
%Determine the italic correction:
    \setbox\z@\hbox{$\macc@style{\macc@nucleus}_{}$}%
    \setbox\tw@\hbox{$\macc@style{\macc@nucleus}{}_{}$}%
    \dimen@\wd\tw@
    \advance\dimen@-\wd\z@
%Now \dimen@ is the italic correction of the symbol.
    \divide\dimen@ 3
    \@tempdima\wd\tw@
    \advance\@tempdima-\scriptspace
%Now \@tempdima is the width of the symbol.
    \divide\@tempdima 10
    \advance\dimen@-\@tempdima
%Now \dimen@ = (italic correction / 3) - (Breite / 10)
    \ifdim\dimen@&gt;\z@ \dimen@0pt\fi
%The bar will be shortened in the case \dimen@&lt;0 !
    \rel@kern{0.6}\kern-\dimen@
    \if#31
      \overline{\rel@kern{-0.6}\kern\dimen@\macc@nucleus\rel@kern{0.4}\kern\dimen@}%
      \advance\dimen@0.4\dimexpr\macc@kerna
%Place the combined final kern (-\dimen@) if it is &gt;0 or if a superscript follows:
      \let\final@kern#2%
      \ifdim\dimen@&lt;\z@ \let\final@kern1\fi
      \if\final@kern1 \kern-\dimen@\fi
    \else
      \overline{\rel@kern{-0.6}\kern\dimen@#1}%
    \fi
  }%
  \macc@depth\@ne
  \let\math@bgroup\@empty \let\math@egroup\macc@set@skewchar
  \mathsurround\z@ \frozen@everymath{\mathgroup\macc@group\relax}%
  \macc@set@skewchar\relax
  \let\mathaccentV\macc@nested@a
%The following initialises \macc@kerna and calls \mathaccent:
  \if#31
    \macc@nested@a\relax111{#1}%
  \else
%If the argument consists of more than one symbol, and if the first token is
%a letter, use that letter for the computations:
    \def\gobble@till@marker##1\endmarker{}%
    \futurelet\first@char\gobble@till@marker#1\endmarker
    \ifcat\noexpand\first@char A\else
      \def\first@char{}%
    \fi
    \macc@nested@a\relax111{\first@char}%
  \fi
  \endgroup
}
\makeatother

\let\oldbar\bar
\renewcommand*{\bar}[1]{{\mathchoice{\widebar{#1}}{\widebar{#1}}{\widebar{#1}}{\oldbar{#1}}}}

% /!\ Il y a parfois des soucis avec cette macro lorsqu'il y a des caractÃ¨res en minuscule en subscript (et dans les captions). Dans ce cas, faire avec \overline et des espaces manuels.

%%% Double bar
% \overline{\overline{\!a}}

%\newcommand\bx{\,\overline{\!x}}
%\newcommand\brho{\,\overline{\!\rho}}
%better widebar a apparemment des soucis avec les \subcaptions pour les CR :shrug:

%%%---------------------------------------------------------------------------

%%% DELIMITERS

%\usepackage{mathtools}
\DeclarePairedDelimiter{\parens}{\lparen}{\rparen}
\DeclarePairedDelimiter{\braces}{\{}{\}}
\DeclarePairedDelimiter{\abs}{\lvert}{\rvert}
\DeclarePairedDelimiter{\norm}{\lVert}{\rVert}
\DeclarePairedDelimiter{\bracks}{[}{]}
\DeclarePairedDelimiter{\angles}{\langle}{\rangle}
\DeclarePairedDelimiter{\aangles}{\langle\!\langle}{\rangle\!\rangle}
\DeclarePairedDelimiter{\pparens}{(\!(}{)\!)}

%%%---------------------------------------------------------------------------

%%% VERTICAL BAR

%%% ''Evaluated at'' macro
\newcommand\restr[2]{{				% we make the whole thing an ordinary symbol
 \left.\kern-\nulldelimiterspace 	% automatically resize the bar with \right
 #1 									% the function
 \vphantom{\big\vert} 				% pretend it's a little taller at normal size
 \right\vert_{#2} 					% this is the delimiter
 }}
% As a subscript: \phi_{F_f\rvert_{K_\ell}} [NO : \phi_{\left.h\right\vert_{K_\ell}}].
% In textstyle : \bigr\rvert_{}.

%%%---------------------------------------------------------------------------

%%% Changer la largeur des adresses

% Jouer avec \hsize :
\makeatletter
\def\@setafterauthor{%
  \vglue3mm%
%  \hspace*{0pt}%
\begingroup\hsize=11.8cm\advance\hsize\abstractmarginL\raggedright
\noindent
%\hspace*{\abstractmarginL}\begin{minipage}[t]{10cm}
   \leftskip\abstractmarginL
  \normalfont\Small
  \@afterauthor\par
\endgroup
\vskip2pt plus 3pt minus 1pt
}
\makeatother

%%%---------------------------------------------------------------------------

\graphicspath{{./figures/}}

\newcommand*{\mk}{\mkern -1mu}
\newcommand*{\Mk}{\mkern -2mu}
\newcommand*{\mK}{\mkern 1mu}
\newcommand*{\MK}{\mkern 2mu}

\hypersetup{urlcolor=purple, linkcolor=blue, citecolor=red}

\newcommand*{\relabel}{\renewcommand{\labelenumi}{(\theenumi)}}
\newcommand*{\romanenumi}{\renewcommand*{\theenumi}{\roman{enumi}}\relabel}
\newcommand*{\Romanenumi}{\renewcommand*{\theenumi}{\Roman{enumi}}\relabel}
\newcommand*{\alphenumi}{\renewcommand*{\theenumi}{\alph{enumi}}\relabel}
\newcommand*{\Alphenumi}{\renewcommand*{\theenumi}{\Alph{enumi}}\relabel}
\let\oldtilde\tilde
\renewcommand*{\tilde}[1]{\mathchoice{\widetilde{#1}}{\widetilde{#1}}{\oldtilde{#1}}{\oldtilde{#1}}}
\let\oldhat\hat
\renewcommand*{\hat}[1]{\mathchoice{\widehat{#1}}{\widehat{#1}}{\oldhat{#1}}{\oldhat{#1}}}

\title{Toric polar maps, normal crossings and mixed volumes}
\alttitle{Applications polaires toriques, croisements normaux et volumes mixtes}

\author{\firstname{Thiago} \lastname{Fassarella}}
\address{Universidade Federal Fluminense, Instituto de Matem{\'a}tica e Estat{\'i}stica, rua Alexandre Moura 8, S{\~a}o Domingos, 24210-200 Niter{\'o}i RJ, Brazil}
\email{tfassarella@id.uff.br}

\author{\firstname{Nivaldo} \lastname{Medeiros}}
\address[1]{Universidade Federal Fluminense, Instituto de Matem{\'a}tica e Estat{\'i}stica, rua Alexandre Moura 8, S{\~a}o Domingos, 24210-200 Niter{\'o}i RJ, Brazil}
\email{nivaldomedeiros@id.uff.br}

\author{\firstname{Rodrigo} \lastname{Salom{\~a}o}}
\address[1]{Universidade Federal Fluminense, Instituto de Matem{\'a}tica e Estat{\'i}stica, rua Alexandre Moura 8, S{\~a}o Domingos, 24210-200 Niter{\'o}i RJ, Brazil}
\email{rsalomao@id.uff.br}

\begin{abstract}
Given a hypersurface in a complex projective space, we prove that the Chern--Schwartz--MacPherson class of a certain open set, namely the complement of the union of the hypersurface and the coordinate hyperplanes, is given (up to sign) by the multidegrees of associated toric polar map, in two particular cases: normal crossings with the coordinate hyperplanes and a nondegeneracy condition with respect to the Newton polytope. In the latter case, we also recover the multidegrees from mixed volumes.
\end{abstract}

\begin{altabstract}
\'Etant donnÃ© une hypersurface dans un espace projectif complexe, nous prouvons que la classe de Chern--Schwartz--MacPherson d'un certain ensemble ouvert, Ã&nbsp; savoir le complÃ©ment de l'union de l'hypersurface et des hyperplans coordonnÃ©s, est donnÃ©e (Ã&nbsp; un signe prÃ¨s) par les multidegrÃ©s de l'application polaire torique associÃ©e, dans deux cas particuliers : les croisements normaux avec les hyperplans coordonnÃ©s et une condition de non dÃ©gÃ©nÃ©rescence par rapport au polytope de Newton. Dans ce dernier cas, nous rÃ©cupÃ©rons Ã©galement les multidegrÃ©s Ã&nbsp; partir de volumes mixtes.
\end{altabstract}

\begin{document}

\maketitle

\section{Introduction}

To each homogeneous polynomial $f \in \C[x_0,\dotsc,x_n]$, we associate the rational map defined by its ``logarithmic derivatives'', which we call the \emph{toric polar map} of $f$, namely $ T_f\colon \P^{n} \dashrightarrow \P^{n}$ given by $x\mapsto \parens[\big]{x_{0}f_{x_0}(x):\dotsb :x_{n}f_{x_n}(x)}. $ Here, $x=(x_0:\dotsb:x_n)$ and $f_{x_i}$ stands for the partial derivative. Let $D=V(f)\subset \P^n$ be the hypersurface given by the zero locus scheme of $f$ and let $\Hcal=V(x_0\dotsm x_n)\subset\P^n$ be the union of the $n+1$ coordinate hyperplanes. We consider a distinguished open set, namely the complement of the union of the hypersurface and the coordinate hyperplanes,
\(
\mathbb \P^{n}\setminus(D\cup\Hcal),
\) which we call the \emph{standard complement of $D$}.

The main theme of this short note is to establish that, up to sign, the \emph{Chern--Schwartz--MacPherson} (CSM) class of the standard complement is given by the multidegrees of the toric polar map assuming that the hypersurface is ``sufficiently general'':
\begin{theo}\label{thm-A}
Let $f\in \C[x_0,\dotsc,x_n]$ be a homogeneous polynomial and let $D=V(f)\subset\P^n$ be the associated hypersurface. Let $d_0,\dotsc,d_n$ be the multidegrees of the toric polar map $T_f$. Assume that either:
\begin{enumerate}\alphenumi
\item \label{thm-A_a} $D\cup\Hcal$ has normal crossings along $D\cap\Hcal$.
\item \label{thm-A_b} The polynomial $f$ is nondegenerate with respect to its Newton polytope.
\end{enumerate}
Then
\begin{equation}\label{eq-CSM}
\csm\parens[\big]{\P^n\setminus (D\cup\Hcal)} = \sum_{i=0}^{n} (-1)^{i} d_{i}\,[\P^{n-i}]
\quad \in A_*\P^n.
\end{equation}
\end{theo}

The relevant definitions are recalled in Section~\ref{section-preliminaries}. Proofs of cases~\eqref{thm-A_a} and~\eqref{thm-A_b} of Theorem~\ref{thm-A} are given in Theorem~\ref{thm-normal-crossings} and Corollary~\ref{cor-fnondegHuh}, respectively.

This note may be seen as an announcement for our forthcoming~\cite{FMS23}, where we prove that identity~\eqref{eq-CSM} holds for any hypersurface. The techniques we employ here are considerably different from the ones in~\cite{FMS23}. As a payoff for considering particular cases, we have more precise information about the geometry of the base locus.

The study of the toric polar map is connected with quite a few important applications, such as geometric modeling~\cite{GPS10}, algebraic statistics and the maximum likelihood degree~\cite{Huh13}, $A$-discriminantal hypersurfaces~\cite{Kap91} and hyperplane arrangements~\cite[Remark~2.6]{FMS23}. Some results on Cremona monomial transformations have been obtained by the first two authors~\cite{FM22}. There is also a relationship with toric varieties, which is the setting of our proof for the second case of Theorem~\ref{thm-A}: here, by using ideas of A.~Khovanskii~\cite{Kh77} and a good resolution of singularities, we are able to relate the multidegrees of the toric polar map with mixed volumes (Theorem~\ref{thm-nondegenerate}), from which the connection with CSM classes follows by~\cite{Huh13}.

From a different perspective, the toric polar map shares many analogies with the classical \emph{gradient map} $\grad f\colon \P^n\dashrightarrow\P^n$, given by the partial derivatives of $f$. The sequence of multidegrees of the later is of fundamental importance in many applications. For instance, J.~Huh~\cite{Huh12} used the fact that this sequence is log-concave to prove Read's unimodality conjecture and more generally Rota--Welsh's conjecture. As another example, P.~Aluffi proved a beautiful formula (recalled in Theorem~\ref{teo-keycsm} below), relating that sequence with the CSM class of $D$. Our formula~\eqref{eq-CSM} can be seen as a version of Aluffi's for a \emph{very affine} hypersurface $D^*\coloneqq D\setminus \Hcal$ inside the torus $(\C^*)^n$: indeed, since $\csm(\P^n)-\csm(\Hcal)=[\P^n]$, it follows from~\eqref{eq-CSM} that
\begin{equation}\label{eq-csm-D*}
\csm(D^*)=\sum_{i=1}^{n} (-1)^{i+1} d_i[\P^{n-i}] \quad \in A_*\P^n.
\end{equation}
Since the degree of the CSM class is the topological Euler characteristic, we obtain from~\eqref{eq-csm-D*} that $\euler(D^*)=(-1)^{n+1} d_n$, where $d_n$ is the topological degree of $T_f$. A similar result has been proved by Dimca--Papadima~\cite{DP03}, relating the topological degree of $\grad f$ with the Euler characteristic of complement of a hypersurface in affine space, which in turn was used to prove a conjecture of I.~Dolgachev~\cite{Dol00}.

It follows from~\eqref{eq-csm-D*} that the coefficients of the \emph{signed} class $(-1)^{n-1}\check{c}_{\SM}(D^*)=\sum_{i=1}^{n} d_i[\P^{n-i}]$ are nonnegative. The positivity of Euler characteristics and CSM classes is a significant and current topic of interest, see e.g.~\cite{EGM18, AMSS22, AMSS23}.

For a more comprehensive list of connections and references, we refer the reader to the introduction in our~\cite{FMS23}.

Let us describe the contents of the paper.

In Section~\ref{section-preliminaries} we setup our notation, provide a quick review of CSM classes and gather related results concerning multidegrees and Segre classes. In Section~\ref{section-normal-crossings}, we prove Theorem~\ref{thm-A} when $D\cup\Hcal$ has normal crossings along $D\cap\Hcal$, see Theorem~\ref{thm-normal-crossings}. In Section~\ref{section-mixed-volumes}, we prove Theorem~\ref{thm-A} holds for a nondegenerate polynomial. The result is obtained as a consequence of Theorem~\ref{thm-nondegenerate}, where we prove that the multidegrees of the toric polar map can be computed directly from the Newton polytope via mixed volumes.

\section{Setup}\label{section-preliminaries}

Let $f\in \C[x_0,\dotsc,x_n]$ be a homogeneous polynomial of degree $k\ge 1$ and let $T_f\colon \P^n\dashrightarrow \P^n$, given by $x \mapsto \parens[\big]{x_{0}f_{x_{0}}(x):\dotsb :x_{n}f_{x_{n}}(x)}$, be the associated toric polar map. Let $\toricscheme(D)$ be the scheme defined by the ideal $\angles{x_0f_{x_0},\dotsc,x_n f_{x_n}}$, which we call the \emph{toric Jacobian ideal} of $D$.

We may always assume $\gcd(f,x_0\dotsm x_n)=1$. Indeed, if $g=x_if$, then an argument with Euler's formula shows that the maps $T_f$ and $T_g$ differ by a linear change of coordinates, so they are defined by the same linear system.

\subsection*{Multidegrees}

Let $\Gamma \subset \P^n\times \P^n$ denote the graph of $T_f$. Let $h$ (resp. $k$) be the pullback of the hyperplane class from the first (resp. second) factor. The class of $\Gamma$ in the Chow ring of $ \P^n\times \P^n$ determines integers $d_0, \dotsc, d_n$ such that
\[\label{eq-graus-proj}
[\Gamma] = d_0 k^n+d_1k^{n-1}h+\dots +d_nh^n
\quad \in A_n(\P^n\times \P^n).
\]
These integers are called the \emph{multidegrees} of $T_f$. If $\mathbb P^{n-j}\subset \P^n$ is a general linear subspace of codimension~$0\leq j \leq n$, then $d_j$ is the degree of the closed subset $\overline{T_f^{-1}(\P^{n-j})}$. In particular, $d_0=1$ and $d_n$ is the (\emph{topological}) \emph{degree of $T_f$}.

As a final remark, suppose $\gcd(f,x_0\dotsm x_n)=1$. If $f$ is reduced, then the polynomials $x_i f_{x_i}$ are relatively prime, so the base locus of $T_f$ agrees with $\toricscheme(D)$. In general, for $f=\Pi_{i=1}^l\, f_i^{n_i}$ an irreducible factorization, we may write the toric polar map as $\bar{T}_f=(x_0 f_{x_0}/g:\dotsb:x_n f_{x_n}/g)$ where $g=\Pi_{i=1}^l\, f_i^{n_i-1}= \gcd(x_0f_{x_0},\dotsc,x_nf_{x_n})$. So, $\toricscheme(D)$ may differ from the base locus of ${T}_f$. In particular,
\[
d_1(T_f)=\deg(x_i{f_{x_i}}/{g}) = \deg(f_\mathrm{red})
\]
where $f_\mathrm{red}=f_1\dotsb f_l$.

\subsection*{CSM classes}

Among the many possibly notions of Chern classes for a singular variety $X$, the
\emph{Chern--Schwartz--MacPherson} (CSM) class $\csm(X)$, in the Chow group $A_*X$, will be particularly important to us. These classes are defined for arbitrary varieties and, for smooth varieties, they agree with the Chern class of the tangent bundle. The degree of the CSM class of a variety is its topological Euler characteristic, and the CSM class satisfy inclusion-exclusion; so, these classes may be seen as a generalization of the Euler characteristic.

We refer to~\cite{AluCSMintro} for a gentle introduction. For a comprehensive survey and detailed account on the history of characteristic classes, we recommend~\cite{Brasselet33CM} and~\cite{Handbook-Sings}. For a quick summary of their basic properties, see also~\cite[Section~2.1]{Alu19} or~\cite[Section~3]{FMS23}.

\subsection*{Aluffi's operators}

Define two linear operators in $A_*X$ as follows. Given a rational equivalence class $\alpha=\bigoplus_i a^i\in A_*X$, indexed by codimension, and a line bundle $\Lcal$ on $X$, set
\[
\alpha^\vee\coloneqq \sum_i (-1)^i a^i
\qquad\text{and}\qquad
\alpha\otimes_X \Lcal \coloneqq \sum_{i\geq 0} \frac{a^i}{c(\Lcal)^i}.
\]
The subscript to $\otimes$ indicates the ambient in which the codimension is taken and we will omit it if there is no risk of confusion. By abuse of language we write $\alpha\otimes \Lcal$ for $\alpha\in A_*Y$ if $Y\subseteq X$. We refer to~\cite[Section~2]{Alu94} and~\cite{AluCSMintro} for further information.

In terms of the above notation we can state a fundamental result due to P.~Aluffi, relating the CSM class of a hypersurface and its singular locus:
\begin{theo}[{\cite[Theorem~I.4]{Alu99a}}]\label{teo-keycsm}
Let $D$ be a hypersurface in a nonsingular complete variety $M$. Then
\[\label{eq-alu99a}
\csm(D)=c(TM)\cap \parens[\Bigg]{\frac{[D]}{1+D}+c\parens[\big]{\Ocal(D)}^{-1} \cap \parens[\bigg]{s\parens[\big]{\sing(D),M}^\vee
\otimes_{M} \Ocal(D)}} \quad \in A_*M.
\]
\end{theo}

Here, $\sing(D)$ is the singular locus scheme of $D$ and $s(X,M) \in A_*X$ is the Segre class (see~\cite[Section~4.2]{Ful84}). Theorem~\ref{teo-keycsm} is the source of many applications involving the Chern--Schwartz--MacPherson class.

Now we present the well known relationship between multidegrees and Segre classes. The statement below is~\cite[Proposition~3.1]{Alu03}, slightly adapted for our needs. Let $h=c_1\parens[\big]{\Ocal(1)}\in A_*\P^n$ denote the hyperplane class.
\begin{prop}\label{prop-segre-proj-degs}
Given an ideal $I=\angles{g_0,\dotsc,g_n}\subset\C[x_0,\dotsc,x_n]$ defined by polynomials of the same degree $r$, let $\varphi_I\colon \P^n\dashrightarrow\P^n$ be the rational map defined by the ratio $(g_0:\dotsb:g_n)$. Let $\Bcal_I$ be the scheme defined by $I$. Then the multidegrees $d_0,\dotsc,d_n$ of $\varphi_I$ can be computed from the Segre class $s(\Bcal_I,\P^n)$ as follows:
\begin{equation}\label{eq-graf-seg}
\sum_{i\geq 0} (-1)^id_ih^i = c\parens[\big]{\Ocal(rh)}^{-1} \cap \parens[\big]{1-s(\Bcal_I,\P^n)^\vee\otimes_{\P^n} \Ocal(rh)} \quad \in A_{*}\P^{n}.
\end{equation}
\end{prop}

\section{Normal crossings}\label{section-normal-crossings}

Recall that a hypersurface $E\subset \P^n$ has \emph{normal crossings} at a point $p$ if its irreducible components passing through $p$ intersect there transversally; that is, if $V(f_1), \dotsc, V(f_r)$ are those components, then $f_1, \dotsc, f_r$ form a regular system of parameters at $p$. We say that $E$ has \emph{normal crossings along} $F\subset E$ if it has normal crossings at any point $p\in F$.

\begin{theo}\label{thm-normal-crossings}
Let $d_0,\dotsc,d_n$ be the multidegrees of the toric polar map $T_{f}$. If ${D}\cup \Hcal$ has normal crossings along $D\cap\Hcal$, then
\[
\csm\parens[\big]{\P^n\setminus (D\cup\Hcal)} = \sum_{i=0}^{n} (-1)^{i} d_{i} h^i
\quad \in A_{*}\mathbb P^{n}.
\]
\end{theo}

The idea is the following. Let $\toricscheme(D)$ be the scheme defined by the homogeneous ideal $\angles{x_0 f_{x_0},\dotsc,x_n f_{x_n}}$. The normal crossing hypothesis implies that the singular locus of $D$ agrees with $\toricscheme(D)$ (see Lemma~\ref{lemma-baselocus}). So we may replace $s\parens[\big]{\sing(D),\P^n}$ by the Segre class $s\parens[\big]{\toricscheme(D),\P^n}$ in Theorem~\ref{eq-alu99a} in order to compute $\csm(D)$; that, combined with Proposition~\ref{prop-segre-proj-degs}, yields $\csm(D)$ in terms of the multidegrees of $T_f$. Finally, again under the normal crossings assumption, our Lemma~\ref{lemma-union-general-section} gives a formula for $\csm\parens[\big]{\P^n\setminus (D\cup\Hcal)}$ in terms of $\csm(D)$ and we are done.

We will prove Theorem~\ref{thm-normal-crossings} at the end of this section. Here is an example that shows concretely the inner workings of the proof.

\begin{exem}\label{example-two-hypersurfaces}
We consider the simplest reducible case. Let $D\subset \P^n$ be union of two smooth hypersurfaces $A, B$ of degrees $a, b$. Assume $D\cup\Hcal$ has only normal crossings. Let us prove Theorem~\ref{thm-normal-crossings} in this case.

The singular locus $Y\coloneqq \sing(D)$ is a smooth complete intersection, so we may use the adjunction formula~\cite[Example~3.2.12]{Ful84} to compute its Segre class; we obtain
\[
s(Y,\P^n) = s(N_{Y/\P^n})\cap [Y] = \frac{[Y]}{(1+A)(1+B)}.
\]
Now $[Y]=abh^2\in A_*\P^n$, and thus
\begin{equation}\label{sex}
s(Y,\P^n) = \frac{abh^2}{(1+ah)(1+bh)} \eqqcolon\sum_{i\geq 0} s_ih^i
\quad \in A_*\P^n.
\end{equation}
Let us relate this class with the multidegrees $d_0,\dotsc,d_n$ of $T_f$. By Lemma~\ref{lemma-baselocus} below, the scheme $\toricscheme(D)$ agrees with $Y$. Replacing $s\parens[\big]{\toricscheme(D),\P^n}=s(Y,\P^n)$ and parsing all symbols, equation~\eqref{eq-graf-seg} reads
\[
\begin{split}
\sum_{i\geq 0} (-1)^id_i h^i 
&amp; = \frac{1}{1+rh} \parens[\Bigg]{1- \sum_{i\geq 0} \frac{(-1)^is_ih^i}{(1+rh)^i}} \\
&amp; = \frac{1}{1+(a+b)h} \parens[\bigg]{1- \sum_{i\geq 0} s_iz^i}
\quad \in A_*\P^n
\end{split}
\]
where $r=\deg D = a+b$ is the degree of the polynomials defining the toric polar map, and where we set $z\coloneqq -h/\parens[\big]{1+(a+b)h}$. But here we recognize the series $\sum s_iz^i$: our expression for the Segre class in~\eqref{sex} yields $\sum s_iz^i=abz^2/(1+az)(1+bz)$ and hence
\[
\sum_{i\geq 0} (-1)^id_i h^i =
\frac{1}{1+(a+b)h} \parens[\Bigg]{1- \frac{abz^2}{(1+az)(1+bz)}}
\quad \in A_*\P^n.
\]
On the other hand, since $D\cup\Hcal$ has normal crossings, it follows from the proof of~\cite[Theorem~1]{Alu99b} (see also Lemma~\ref{lemma-union-general-section} below) that the CSM class of its complement is also given in terms of $a$ and $b$, namely
\[
\csm\parens[\big]{\P^n\setminus(D\cup\Hcal)} =
\frac{(1+h)^{n+1}}{(1+ah)(1+bh)\prod_{i=0}^n(1+h)} =
\frac{1}{(1+ah)(1+bh)}
\quad \in A_*\P^n.
\]
Therefore, in order to prove that $\csm\parens[\big]{\P^n\setminus (D\cup\Hcal)}=\sum(-1)^id_ih^i$ as stated in Theorem~\ref{thm-normal-crossings}, we need only to verify that the formal identity
\[
\frac{1}{(1+ah)(1+bh)}=\frac{1}{1+(a+b)h} \parens[\Bigg]{1- \frac{abz^2}{(1+az)(1+bz)}}
\]
(recall $z=-h/\parens[\big]{1+(a+b)h}$) holds, which we leave as an exercise to the reader.
\end{exem}
Now we head towards the proof of Theorem~\ref{thm-normal-crossings}. We need two auxiliary results.

\begin{lemm}\label{lemma-baselocus}
Let $\toricscheme(D)$ be the scheme given by the ideal $\angles{x_0f_{x_0},\dotsc,x_nf_{x_n}}$. If ${D}\cup\Hcal$ has normal crossings along ${D}\cap\Hcal$, then the scheme $\toricscheme(D)$ coincides with $\sing(D)$.
\end{lemm}

\begin{proof}
We consider the affine chart $\mathbb{A}^n$ given by $x_0\neq 0$ and let $B, J \subset\C[x_1,\dotsc,x_n]$ be the dehomogenization of the toric Jacobian ideal $\angles{x_0f_{x_0},\dotsc,x_nf_{x_n}}$ and the Jacobian ideal $\angles{f_{x_0},\dotsc,f_{x_n}}$, respectively. By Euler's identity, we may replace $f_{x_0}$ by $f$, so that
\[
B = \angles{ f, x_1f_{x_1},\dotsc,x_nf_{x_n} }
\qquad\text{and}\qquad
J = \angles{ f, f_{x_1},\dotsc,f_{x_n} }.
\]
Note that the normal crossings hypothesis implies that $\gcd(f,x_1\dotsm x_n)=1$. Clearly we have $B\subseteq J$ and our aim is to prove that for each $p\in \mathbb{A}^n$ we have
\begin{equation}\label{eq-cruznorm-1}
B_p = J_p \quad \subseteq \Ocal_p=\Ocal_p(\mathbb{A}^n),
\end{equation}
which is enough to prove the lemma.

Let $p \in \A^n$. Note that if $p\not\in D$ (so that $f$ is invertible) or $p\not\in\Hcal$ (all $x_i$ are invertible) then the equality $B_p=J_p$ holds trivially, so we may assume $p\in D\cap\Hcal$. Since ${D}\cup \Hcal$ has normal crossings on ${D}\cap\Hcal$, after eventually relabeling indexes we may write
\[
f_1,\dotsc,f_r
\qquad\text{and}\qquad
x_s,\dotsc,x_n
\]
to denote the irreducible components of ${D}$ and $\Hcal$ passing through $p$, where $1\leq r &lt; s\leq n$. Since $x_1,\dotsc,x_{s-1}$ are invertible in $\Ocal_p$, we have
\begin{equation}\label{eq-cruznorm-2}
B_p = \ \angles{f,f_{x_1},\dotsc,f_{x_{s-1}}, x_sf_{x_s},\dotsc,x_nf_{x_n}}
\qquad\text{and}\qquad
J_p = \ \angles{ f, f_{x_1},\dotsc,f_{x_n} }.
\end{equation}

Claim: \emph{We may discard all the components of $D$ that do not pass through $p$.}

Indeed, write $f=f_1 \dotsm f_r\cdot g$ with $g(p)\neq 0$. Define $\tilde{f}=f_1 \dotsb f_r$ so that $f=\tilde{f}\cdot g$. Then $g$ is invertible in $\Ocal_p$ and
\[
f_{x_j}=\tilde{f}_{x_j}g + g_{x_j}\tilde{f}
\quad\text{for $j=1,\dotsc,n$}.
\]
Now, let $\tilde{B}_p$ and $\tilde{J}_p$ be ideals defined as in~\eqref{eq-cruznorm-2} but with $\tilde{f}$ replacing $f$:
\[
\tilde{B}_p=\angles{\tilde{f},\tilde{f}_{x_1},\dotsc,\tilde{f}_{x_{s-1}}, x_s\tilde{f}_{x_s},\dotsc,x_n\tilde{f}_{x_n}}
\qquad\text{and}\qquad
\tilde{J}_p=\angles{ \tilde{f}, \tilde{f}_{x_1},\dotsc,\tilde{f}_{x_n} }.
\]
Hence $B_p=\tilde{B}_p$ and $J_p=\tilde{J}_p$, so we may replace $f$ by $\tilde{f}$, as claimed.

So, assume that $f=f_1 \dotsm f_r$, with $f_i$ irreducible and $f_i(p)=x_j(p)=0$ for $1\leq i \leq r &lt; s\leq j\leq n$. Write the partial derivatives of $f$ in a more convenient way:
\begin{equation}\label{eq-cruznorm-3}
{f}_{x_\ell} = \hf_1 \dv{f_1}{x_\ell} + \hf_2 \dv{f_2}{x_\ell} + \dotsb + \hf_r\dv{f_r}{x_\ell}
\quad (\ell=1,\dotsc,n)
\end{equation}
where $\hf_i\coloneqq 
\frac{f}{f_i}$. Now we consider the ideal $I_p\coloneqq \angles{\hf_1,\dotsc,\hf_r}$. From~\eqref{eq-cruznorm-3} we get inclusions
\[
B_p\subseteq J_p\subseteq I_p.
\]
We shall prove that $I_p\subseteq B_p$ so the three ideals coincide, therefore proving~\eqref{eq-cruznorm-1}.

Since ${D}\cup \Hcal$ has normal crossings at $p$, we have
\[
(\dd f_1 \wedge \dotsb \wedge \dd f_r \wedge \dd x_{s} \wedge \dotsb \wedge \dd x_n)(p)\neq 0
\]
that is, the $(r+n-s+1)\times n$ matrix
\[
{\left[
\begin{array}{ccc|ccc}
\dv{f_1}{x_1} &amp;\dotsb &amp; \dv{f_1}{x_{s-1}} &amp; \dv{f_1}{x_{s}} &amp; \dotsb &amp; \dv{f_1}{x_n} \\
\vdots &amp; &amp; \vdots &amp; \vdots &amp; &amp; \vdots \\
\dv{f_r}{x_1} &amp;\dotsb &amp; \dv{f_r}{x_{s-1}} &amp; \dv{f_r}{x_{s}}&amp;\dotsb &amp; \dv{f_r}{x_{n}} \\[0.25em]
\hline &amp; &amp; &amp; {} &amp; &amp; \\
&amp; &amp; &amp; &amp;\mathbf{I_{n-s+1}} &amp; \\
&amp;&amp; &amp; &amp;&amp; {}
\end{array}
\right]}(p)
\]
has maximal rank $r+n-s+1$. Thus, on the top left block, there is a $r\times r$ submatrix $A$ which is invertible in $\Ocal_p$, say with columns $j_1,\dotsc,j_r$. But its transpose $A^t$ already appeared implicitly in~\eqref{eq-cruznorm-3}:
\[
\begin{bmatrix}
{f}_{x_{j_1}} \\
\vdots\\
{f}_{x_{j_r}}
\end{bmatrix}
=
\begin{bmatrix}
\dv{f_1}{x_{j_1}} 	&amp; \dotsb 	&amp; \dv{f_r}{x_{j_1}}\\
\vdots 				&amp; 			&amp; \vdots \\
\dv{f_1}{x_{j_r}} 	&amp; \dotsb 	&amp; \dv{f_r}{x_{j_r}}
\end{bmatrix}
\begin{bmatrix}
\hf_1	\\
\vdots	\\
\hf_r
\end{bmatrix}.
\]
Hence $\angles{\hf_1,\dotsc,\hf_r}\subseteq \angles{{f}_{x_{j_1}},\dotsc,{f}_{x_{j_r}}}$, which yields $I_p\subseteq B_p$, as required.
\end{proof}

\begin{lemm}\label{lemma-union-general-section}
Let $D\subset\P^n$ be a hypersurface. Assume that $D\cup \Hcal$ has normal crossings along $D\cap
\Hcal$. Then
\[
\csm({\P^n\setminus D}) = (1+h)^{n+1} \cdot \csm\parens[\big]{\P^n\setminus(D\cup\Hcal)}
\quad \in A_*\P^n.
\]
\end{lemm}
\begin{proof}
Let $C\subset \sing(D)$ be the set of points where $D$ does not have normal crossings. Then $C$ cannot meet $\Hcal$, so $C$ is finite. Take $\pi\colon M\to\P^n$ a proper birational map, with $M$ a nonsingular variety, so that {$\pi^{-1}({D}\cup\Hcal)_\mathrm{red}$} is a divisor with smooth components and normal crossings, and \smash{$\restr{\pi}{M\setminus \pi^{-1}(C)}$} is an isomorphism. Set $D'=\pi^{-1}(D)_{\mathrm{red}}$ with components ${D'_j}$, let $\tilde{H}_i$ be the strict transforms of the coordinate hyperplanes $H_i$ and put \smash{$\tilde \Hcal=\sum \tilde{H}_i$}. Since we have smooth components with normal crossings, from the proof of~\cite[Theorem~1]{Alu99b} and after some simplifications, we get
\[\label{eq-naosei2}
\csm({M\setminus {D'}}) = \prod_{i=0}^n(1+\tilde{H}_i) \cdot \csm\parens[\big]{M\setminus({D'}\cup\tilde\Hcal)}
\]
and then the conclusion follows by functoriality of the CSM class.
\end{proof}

\subsubsection*{Proof of Theorem~\ref{thm-normal-crossings}}

Set $G=\sum_{i\geq 0} (-1)^id_ih^i \in A_{*}\P^{n}$. Let us compute the Chern--Schwartz--MacPherson class of $D$. By Lemma~\ref{lemma-baselocus}, the scheme $\toricscheme(D)$ agrees with the singular locus $\sing(D)$, so in particular $s\parens[\big]{\toricscheme(D),\P^n}=s\parens[\big]{\sing(D),\P^n}.$ Substituting into the formula in Proposition~\ref{prop-segre-proj-degs} yields $ G=c\parens[\big]{\Ocal(kh)}^{-1}\cap[\P^n] - c\parens[\big]{\Ocal(kh)}^{-1}\cap \parens[\big]{s\parens[\big]{\sing(D),\P^n}^\vee \otimes \Ocal(kh)}, $ where $k\coloneqq \deg D$ is the common degree of the polynomials defining $T_f$. Since $\Ocal(kh)=\Ocal(D)$, we get
\[
c\parens[\big]{\Ocal(D)}^{-1} \cap \parens[\bigg]{s\parens[\big]{\sing(D),\P^n}^\vee \otimes \Ocal(D)} =
\frac{1}{1+[D]} - G.
\]
Plugging this expression into Theorem~\ref{teo-keycsm} yields
\[
\csm(D) = {c(T\P^n)\cap}
\parens[\Bigg]{\frac{[D]}{1+D} + {\frac{1}{1+[D]}} -G } = {c(T\P^n)\cap} (1-G).
\]
Hence $\csm({\P^n\setminus D}) = \csm({\P^n})-\csm(D) ={c(T\P^n)\cap}[\P^n] - {c(T\P^n)\cap} (1-G) $ and therefore
\begin{equation}\label{eq-crnormais-2}
\csm({\P^n\setminus D}) = (1+h)^{n+1}\cdot G.
\end{equation}
Finally, by Lemma~\ref{lemma-union-general-section} we have
\[
\csm({\P^n\setminus D}) = (1+h)^{n+1} \cdot \csm\parens[\big]{\P^n\setminus (D\cup\Hcal)}
\]
and hence from~\eqref{eq-crnormais-2} we obtain $\csm\parens[\big]{\P^n\setminus (D\cup\Hcal)}=G$, which is exactly what we needed to prove. This concludes the proof of Theorem~\ref{thm-normal-crossings}.\qed

\section{Mixed volumes}\label{section-mixed-volumes}

Consider a nonzero Laurent polynomial
\[
g=\sum c_{I}x^I \quad \in\C[x_1^{\pm 1},\dots, x_n^{\pm 1}].
\]
Its \emph{Newton polytope} $\Delta_g$ is the convex hull in $\R^n$ of the tuples $I\in \Z^n$ such that $c_I\neq 0$. The \emph{support function} $\psi_{\Delta_g}$ of the polytope is the function on $\R^n$ defined by
\[
\psi_{\Delta_g}(v) = \min\braces[\big]{ \angles{v,u} \;;\; u\in \Delta_g }
\]
where $\angles{v,u}$ denotes the usual scalar product.

For each face $\xi$ of $\Delta_g$, let $g_{\xi}=\sum_{I\in\xi}c_Ix^I$ be the Laurent polynomial restricted to that face. A Laurent polynomial $g$ is said to be \emph{nondegenerate} (with respect to its Newton polytope) if $V(g_{\xi})\subset (\C^*)^{n}$ is nonsingular for every face $\xi$ of $\Delta_g$. In particular, $V(g)\subset (\C^*)^{n}$ is nonsingular. We say that a homogeneous polynomial $f$ in $n+1$ variables is \emph{nondegenerate} if its dehomogenization $g$ with respect to a variable is nondegenerate as a Laurent polynomial in $n$ variables. This does not depend on the chosen variable.

\begin{prop}\label{prop-resolutionTf}
If $f$ is a nondegenerate homogeneous polynomial, then there is a refinement $\Sigma$ of the fan of\/ $\P^{n}$ so that $\pi\colon X_{\Sigma}\to\P^{n}$ is a resolution of indeterminacies of $T_{f}$, that is, $T_{f}\circ \pi \colon X_{\Sigma}\to\P^{n}$ is a morphism.
\end{prop}
\begin{proof}
We may assume $\gcd(f,x_0 \dotsb x_n)=1$ and hence $f$ must be reduced, since otherwise it would have a singular point in $(\C^*)^n$. In particular, the base locus of $T_f$ is given by the ideal \smash{$\angles{x_0f_{x_0}, \dots, x_nf_{x_n}}$}.

We denote by $g=\sum c_{I}x^I$ a dehomogenization of $f$ with respect to a variable. Take a sufficiently fine subdivision $\Sigma$ of the fan of $\P^n$ so that the support function of $\Delta_g$ is piecewise linear, i.e., in a way that $\psi_{\Delta_g}$ is linear on each cone of $\Sigma$. See~\cite[Section~6.1]{CLS}.

Following~\cite[Section~2]{Kh77} (see also~\cite[Theorem~3.4]{Ish07}), the corresponding toric variety $X_{\Sigma}$ is nonsingular and the closure $V=\overline{V(g)}$ on $X_{\Sigma}$ has normal crossings with the boundary divisor $E\coloneqq X_{\Sigma}\setminus (\C^*)^n$. We will show that the existence of indeterminacies of $T_f\circ \pi$ on the boundary divisor $E$ is equivalent to the non-transversality of $V$ with $E$.

Since $X_{\Sigma}$ is nonsingular it can be covered by local charts $U_{\sigma}$, where each $\sigma$ is a cone of $\Sigma$ spanned by a basis for the lattice $\Z^n$. Let $v_1,\dotsc, v_n\in \Z^n$ be the set of generators of a given $\sigma$. If we write $v_j=(a_{j1},\dots, a_{jn})$, we deduce that $\restr{\pi}{U_\sigma}\colon U_\sigma \to \C^n$ has
\[
x_i \colon (y_1,\dots, y_n) \mapsto y_1^{a_{1i}}\dotsb y_n^{a_{ni}}
\]
as coordinate functions, for $i=1,\dotsc, n$. In addition, we note that
\[
\pi^*(g) = \sum c_I y_1^{\angles{v_1,I}}\dotsb y_n^{\angles{v_n,I}}.
\]
From the condition of linearity of the support function $\psi_{\Delta_g}$, we obtain $\psi_{\Delta_g}(v)=\angles{v,I_0}$ for all $v\in \sigma$ and for some suitable vertex $I_0$ of $\Delta_g$. Then we can write
\begin{equation}\label{PB}
g^*\coloneqq \pi^*(g) = y^mh(y)
\end{equation}
where $m=\parens[\big]{\psi_{\Delta_g}(v_1),\dots, \psi_{\Delta_g}(v_n)}$ and $h$ is a polynomial with a nonzero constant term.

We want to show that the linear system defining $T_f\circ \pi$ is free of base points. We note that the linear system of $T_f$ has the following set of generators
\[
\angles{g, x_1g_{x_1},\dots, x_ng_{x_n}}
\]
and we have the following relation
\[
\begin{pmatrix}
y_1g^*_{y_{1}} 	\\
\vdots 			\\
y_n g^*_{y_{n}}
\end{pmatrix}
= A\cdot
\begin{pmatrix}
x_1g_{x_1} \vphantom{y_n g^*_{y_{n}}}	\\
\vdots 									\\
x_ng_{x_n} \vphantom{y_n g^*_{y_{n}}}
\end{pmatrix}
\]
where $A$ is the invertible matrix on which has the coordinates of $v_i$ as rows. Therefore, from~\eqref{PB} we deduce that the pullback of the linear system of $T_f$ by $\pi$ is
\[
\angles{y^mh, y^my_1h_{y_1},\dots, y^my_nh_{y_n}}
\]
and we deduce from this that the linear system of $T_f\circ \pi$ is given by
\begin{equation}\label{LinearSystem}
\angles{h, y_1h_{y_1},\dots, y_nh_{y_n}}.
\end{equation}

For each subset $J=\{j_1,\dotsc,j_s\} \subset \{1,\dots, n\}$, let $H_J=\cap_{j\in J} V(y_j)$ be the intersection of the corresponding boundary divisors. It follows from~\eqref{LinearSystem} that the existence of a base point $p\in H_J$ is equivalent to
\[
h(p)=0
\qquad\text{and}\qquad
(\dd y_{j_1} \wedge \dotsb \wedge \dd y_{j_s} \wedge \dd h) (p) =0.
\]
Therefore the linear system defining $T_f\circ \pi$ is free of base points if and only if $V$ has normal crossings with the boundary divisor $E$. This concludes the proof.
\end{proof}

Our next goal is to give, for nondegenerate homogeneous polynomials, an explicit formula for the multidegrees of the toric polar map in terms of the Newton polytope.

Let $\mv_n(P_1,\dotsc,P_n)$ be the \emph{normalized mixed volume} of $n$ polytopes $P_1,\dots, P_n$ in $\R^n$, defined~as
\[
\mv_n(P_1,\dotsc,P_n) =
\sum_{I\subseteq\{1,\dotsc,n\}} (-1)^{n-\abs{I}}\,\vol_n\parens[\bigg]{\sum_{i\in I} P_i}
\]
where $\sum_{i\in I} P_i$ is the Minkowski sum of polytopes and $\vol_n$ is the {standard} $n$-dimensional volume in $\R^n$. See for example~\cite[Chapter~7]{CLO} for more information. Note that
\[
\mv_n(P,\dotsc,P) = n! \, \vol_n(P)
\]
so that the $n$-dimensional standard simplex $\Delta$ in $\R^n$ has normalized volume~$1$.

Let $f$ be a homogeneous polynomial. If $g$ is the dehomogenization with respect to one variable and $h$ is the dehomogenization respect to another variable, then $\Delta_{h}$ is the image of $\Delta_{g}$ by a volume-preserving transformation of $\R^{n}$. Furthermore, the mixed volume is invariant under volume-preserving transformations (see~\cite[Theorem~4.12]{CLO}). So, by abuse of notation, we shall write $\Delta_{f}$ to denote the Newton polytope of a dehomogenization of $f$ with respect to any given variable.

\begin{theo}\label{thm-nondegenerate}
Let $f$ be a nondegenerate homogeneous polynomial in $n+1$ variables and $\Delta_{f}$ be the Newton polytope of a dehomogenization of $f$. If $d_0,\dotsc,d_n$ are the multidegrees of the toric polar map $T_f$, then
\begin{equation}\label{eqs-dj}
d_j= \mv_n(\underbrace{\Delta,\dotsc,\Delta\vphantom{\Delta_f}}_{n-j},\underbrace{\Delta_f,\dotsc,\Delta_f}_{j})
\quad\text{for $j=0,\dotsc,n$},
\end{equation}
where $\Delta\subset\R^{n}$ is the $n$-standard simplex. In particular,
\(
\deg T_f = n! \, \vol_n(\Delta_f).
\)
\end{theo}

\begin{proof}
We use the toric resolution $\pi\colon X_{\Sigma}\to \P^n$ given by Proposition~\ref{prop-resolutionTf}. Let $ H$ and $ G$ denote the pullback of a general hyperplane in $\P^n$ by $\pi$ and $T_f\circ \pi$, respectively. Then
\begin{equation}\label{eq-dj}
d_j = \int_{X_{\Sigma}} H^{n-j}G^{j}.
\end{equation}

Let $f_{a}=a_0x_0f_{x_0}+\dots + a_nx_nf_{x_n}$ be a general member of the linear system defining $T_{f}$. The divisor $G$ corresponds to the strict transform by $\pi$ of $V(f_{a})$. We write $g_{a}$ (resp. $g$) for the dehomogenization of $f_{a}$ (resp. of $f$) with respect to a variable. The rational function $g_a$ on $X_\Sigma$ gives rise to a divisor
\[
\divisor(g_{a}) = G+ \sum_\rho \ord_{D_\rho} (g_a)\cdot D_\rho= G+\sum_{\rho} \psi_{\Delta_{g_{a}}} (v_\rho) D_{\rho}.
\]
Here, $v_\rho$ is the primitive ray generator of a ray $\rho$ of $\Delta_{g_a}$ and $D_{\rho}$ is the torus invariant prime divisor corresponding to $\rho$. Define
\[
E_{{a}} = \sum_{\rho} - \psi_{\Delta_{g_{a}}} (v_\rho) D_{\rho}.
\]
Then $\divisor(g_a)=G-E_a$ and hence
\[
\int_{X_{\Sigma}}H^{n-j}G^{j} = \int_{X_{\Sigma}}H^{n-j}E_{a}^{j}.
\]
Now, we combine the intersection-theoretic characterization of the mixed volume (see~\cite[Section~5.4]{Ful93}) with equation~\eqref{eq-dj} to obtain
\[
d_{j}=
\int_{X_{\Sigma}}H^{n-j}E_{a}^{j} =
\mv_n(
\underbrace{\Delta,\dotsc,\Delta\vphantom{\Delta_{g_{a}}}}_{n-j},
\underbrace{\Delta_{g_{a}},\dotsc,\Delta_{g_{a}}}_{j})
\]
since $H$ and $E_{a}$ are the divisors corresponding to the polytopes $\Delta$ and $\Delta_{g_a}$, respectively. The proof now follows from {the fact that $g_{a}$ and $g$ have the same Newton polytope for a general $a\in\P^{n}$.}
\end{proof}

It follows from Theorem~\ref{thm-nondegenerate} that almost all homogeneous polynomials with a given Newton polytope yield toric polar maps with the same multidegrees.

Now we prove Theorem~\ref{thm-A} for nondegenerate polynomials. For convenience, here is the statement again.

\begin{coro}\label{cor-fnondegHuh}
Let $f$ be a nondegenerate complex homogeneous polynomial in $n+1$ variables. Let $D=V(f)\subset\P^n$ be the associated hypersurface. If $d_0,\dotsc,d_n$ are the multidegrees of\/ $T_f$, then
\[
\csm \parens[\big]{\P^n\setminus(D\cup\Hcal)} = \sum_{i=0}^{n} (-1)^{i} d_{i}\, [\P^{n-i}] \quad \in A_{*}\mathbb P^{n}.
\]
\end{coro}

\begin{proof}
Consider the algebraic torus $(\C^*)^n=\P^n\setminus \Hcal$ and the very affine variety $X=D\cap (\C^*)^n$. Note that $(\C^*)^n\setminus X = \P^n\setminus (D\cup \Hcal)$. By~\cite[Theorem~4.1]{Huh13}, the coefficients of the class $\csm\parens[\big]{(\C^*)^n\setminus X}\in A_*\P^n$ are given exactly by the mixed volumes that appear in~\eqref{eqs-dj}. Hence the result follows from Theorem~\ref{thm-nondegenerate}.
\end{proof}

\begin{exem}
Consider the pencil of plane conics
\[
f_t = x^2+y^2+xz+yz+ txy \quad (t\in \C).
\]
Dehomogenizing with respect to $z=1$ gives $g_{t}= x^2+y^2+x+y+txy$, whose Newton polygon has $(1,0), (2,0), (0,2), (0,1)$ as vertices. Note that $f_t$ is degenerate if and only if $t=2$, the unique singular member of the pencil.

By Theorem~\ref{thm-nondegenerate} we have $\deg T_{f_{t}}={2! \, \vol_2(\Delta_{g_{t}})}=3$ whenever $t\neq 2$. On the other hand, $f_2=(x+y)(x+y+1)$, so $V(f_2)\cup \Hcal$ does not have normal crossings and we cannot apply Theorem~\ref{thm-normal-crossings} either; by direct computation we get $\deg T_{f_2}=1$, hence a birational map.
\end{exem}

\begin{exem}[Cayley cubic]\label{ex-Cayley}
The polynomial
\[
f_1 = x_0x_1x_2 + x_0x_1x_3 + x_0x_2x_3 + x_1x_2x_3
\]
is an equation for the Cayley cubic in $\P^3$, whose singular locus consists exactly of four simple nodes. It is straightforward to check that $f_1$ is not degenerate and its Newton polytope $\Delta_{f_1}$ is a tetrahedron in $\R^3$ of volume~$1/6$; so by Theorem~\ref{thm-nondegenerate} we have $\deg T_{f_1}=1$. One easily sees that $T_{f_1}$ and the standard Cremona transformation $x\mapsto (x_0^{-1}:\dotsb:x_3^{-1})$ are defined by the same linear system, hence by~\cite{GP} the multidegrees are: $1$, $3$, $3$, $1$.

Let $f_2$ be an equation for the Cayley cubic in general position. Then $V(f_2)\cup \Hcal$ has normal crossings along $V(f_2)\cap \Hcal$. By Theorem~\ref{thm-normal-crossings} we have $\deg T_{f_2}=-\euler\parens[\big]{\P^n\setminus \parens[\big]{V(f_2)\cup\Hcal}}$, which is $23$, as it follows by a calculation using inclusion-exclusion. Since the base locus of $T_{f_2}$ consists only of the four nodes (Lemma~\ref{lemma-baselocus}), its restriction to a general $\P^i\subsetneq \P^3$ is a morphism, so the sequence of multidegrees is $1$, $3$, $9$, $23$.

Finally, consider the polynomial
\[
f_3=x_0^3-x_0x_1^2-x_0x_2^2-x_0x_3^2 + 2x_1x_2x_3
\]
which is yet another equation for the Cayley cubic. This is a degenerate polynomial and the corresponding hypersurface is not in general position, so none of our previous results apply. A computation with \textsc{Macaulay~2}~\cite{M2} shows that the multidegrees of $T_{f_3}$ are $1$, $3$, $6$, $8$. On the other hand, by using P.~Aluffi's ``CSM.m2''~\cite{Alu03} or M.~Helmer and C.~Jost ``CharacteristicClasses.m2''~\cite{Jos15}, one computes the CSM class of $V(x_0x_1x_2x_3 f_3)$ and obtains
\(7[\mathbb P^2] + 12[\mathbb P^0]\), hence the conclusion of Theorem~\ref{thm-A} holds in this case as well. This is expected: in our forthcoming~\cite[Theorem~1.2]{FMS23}, we prove that Theorem~\ref{thm-A} holds without any hypothesis on the hypersurface.
\end{exem}

\section*{Declaration of interests}

The authors do not work for, advise, own shares in, or receive funds from any organization that could benefit from this article, and have declared no affiliations other than their research organizations.

\setlength\bibitemsep{.2\baselineskip}

\printbibliography

\end{document}</pre></body></html>