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\title[Contact loci, nearby cycles of nondegenerate polynomials]{Geometry of nondegenerate polynomials: Motivic nearby cycles and Cohomology of contact loci}

\author{\lastname{L\^e} \firstname{Quy Thuong}}
\address{University of Science, Vietnam National University, Hanoi, 334 Nguyen Trai Street, Thanh Xuan District, Hanoi, Vietnam}
\address{Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka, 560-0043, Japan}
\email{leqthuong@gmail.com}


\author{\lastname{Nguyen} \firstname{Tat Thang}\IsCorresp}
\address{Institute of Mathematics, Vietnam Academy of Science and Technology 18 Hoang Quoc Viet Road, Cau Giay District, Hanoi, Vietnam}
\email{ntthang@math.ac.vn}

\thanks{The second author was partially supported by the Vietnam Academy of Science and Technology under Grant Number ĐLTE00.01/21-22. This work was funded by VinGroup and supported by Vingroup Innovation Foundation (VinIF) under the project code VINIF.2021.DA00030.}

\keywords{\kwd{arc spaces}
\kwd{contact loci}
\kwd{motivic zeta function}
\kwd{motivic Milnor fiber}
\kwd{motivic nearby cycles}
\kwd{Newton polyhedron}
\kwd{nondegeneracy}
\kwd{sheaf cohomology with compact support}}
\subjclass{14B05, 14B07, 14J17, 32S05, 32S30, 32S55}

\begin{abstract}
We study polynomials with complex coefficients which are nondegenerate in two senses, one of Kouchnirenko and the other with respect to its Newton polyhedron, through data on contact loci and motivic nearby cycles. Introducing an explicit description of these quantities we can answer in part the question concerning the motivic nearby cycles of restriction functions in the context of Newton nondegenerate polynomials. Furthermore, in the nondegeneracy in the sense of Kouchnirenko, we give calculations on cohomology groups of the contact loci.
\end{abstract}


\begin{document}
\maketitle


\section{Introduction}\label{sec1}
Let $f$ be a nondegenerate $\mathbb C$-polynomial in the sense of Kouchnirenko (cf. Section~\ref{nondegenerate}) vanishing at the origin $O$ of $\mathbb C^d$. The problem of computing the motivic Milnor fiber $\mathscr S_{f,O}$ in terms of the Newton polyhedron $\Gamma$ of $f$ was early mentioned in the works~\cite{AB-CN-L-MH} and~\cite{G} with materials coming from~\cite{DH} (see also~\cite{GLM2} for a generalization). Recently, Steenbrink and Bultot--Nicaise obtain solutions in terms of toric geometry (\cite{St2}), or of log smooth models (\cite{BN}). Their formula for $\mathscr{S}_{f,O}$ together with the additivity of the Hodge spectrum operator allows to reduce the computation of the Hodge spectrum of $(f,O)$ to that of quasi-homogeneous singularities. In this article, we will show that the formula also provides a way to explore the following problem for Newton nondegenerate polynomials.

\begin{enonce}{Problem}\label{pro1}
Let $f$ be in $\mathbb C[x_1,\dots,x_d]$ with $f(O)=0$, and let $H$ be a linear hyperplane in $\mathbb C^d$. What is the relation between $\mathscr{S}_{f,O}$ and $\mathscr{S}_{f|_H,O}$?
\end{enonce}

The question concerns a motivic analogue of a monodromy relation of a complex singularity and its restriction to a generic hyperplane studied early in~\cite{LDT75,LDT77}. For $n\in \mathbb N^*$, the $n$-iterated contact locus $\mathscr X_{n,O}(f)$ (cf. Section~\ref{section2.3}) admits a decomposition as a disjoint union into its $\mu_n$-invariant $\mathbb C$-subvarieties $\mathscr{X}_{J,a}^{(n)}$ along $a\in (\mathbb N^*)^J$ and $J\subseteq [d]:=\{1,\dots,d\}$. The nondegeneracy of $f$ allows to describe $\mathscr{X}_{J,a}^{(n)}$ via $\Gamma$, as in Theorem~\ref{mainthm1}, which is the key step to compute the motivic zeta function $Z_{f,O}(T)$ and the motivic Milnor fiber $\mathscr S_{f,O}$, which yields a proof of Theorem~\ref{thm4.1}. Note that this theorem is well known as mentioned above (see~\cite{G, AB-CN-L-MH, GLM2}). For every face $\gamma$ of $\Gamma$, let $J_{\gamma}$ be the unique subset of $[d]$ such that $\gamma$ is contained in the hyperplanes $x_j=0$ for all $j\not\in J_{\gamma}$ and not contained in the other coordinate hyperplanes, and let $X_{\gamma}(0)$ (resp. $X_{\gamma}(1)$) be the $\mathbb C$-subvariety of $\mathbb G_{m,\mathbb C}^{J_{\gamma}}$ defined by the face function $f_{\gamma}$ (resp. $f_{\gamma}-1$).

\begin{theo*}[see Theorem~\ref{thm4.1}]
Let $f$ be in $\mathbb C[x_1,\dots,x_d]$ with $O\in X_0:= f^{-1}(0)$, let $d_1$ and $d_2$ be in $\mathbb N$ such that $d=d_1+d_2$. The below hold in $\mathscr{M}_{X_0}^{\hat\mu}$ for \eqref{thi}, in $\mathscr{M}_{\mathbb A_{\mathbb C}^{d_1}}^{\hat\mu}$ for \eqref{thii}, and in $\mathscr{M}_{\mathbb C}^{\hat\mu}$ for \eqref{thiii}.
\begin{enumerate}\romanenumi
\item \label{thi} If\/ $f$ is Newton nondegenerate, then
\begin{align*}
\mathscr{S}_f=-\sum_{\gamma\in F\setminus \widetilde{F}}\lambda_{\gamma}\left[X_{\gamma}(1)\to {X_0}\right]+\sum_{\gamma\in F}\lambda_{\gamma}\left[X_{\gamma}(0)\to {X_0}\right].
\end{align*}
\item \label{thii} If\/ $f$ is Newton nondegenerate and $\iota: \mathbb A_{\mathbb C}^{d_1}\equiv \mathbb A_{\mathbb C}^{d_1}\times_{\mathbb C}\{0\}^{d_2}\hookrightarrow X_0$ is an inclusion, then
\begin{align*}
\iota^*\mathscr{S}_f=-\sum_{\gamma\in F(d_1)\setminus \widetilde{F}}\lambda_{\gamma}\left[X_{\gamma}(1)\times_{X_0}\mathbb A_{\mathbb C}^{d_1}\to \mathbb A_{\mathbb C}^{d_1}\right]+\sum_{\gamma\in F(d_1)}\lambda_{\gamma}\left[X_{\gamma}(0)\times_{X_0}\mathbb A_{\mathbb C}^{d_1}\to \mathbb A_{\mathbb C}^{d_1}\right].
\end{align*}
\item \label{thiii} If\/ $f$ is nondegenerate in the sense of Kouchnirenko, then
\[
\mathscr{S}_{f,O}=\sum_{\gamma\in K}(-1)^{|J_{\gamma}|+1-\dim(\gamma)}\left([X_{\gamma}(1)]-[X_{\gamma}(0)]\right).
\]
Here, $\iota^*$, $F$, $\widetilde{F}$, $F(d_1)$, $K$, and $\lambda_{\gamma}$ are defined in Sections~\ref{sect2.1}, \ref{nondegenerate} and~\ref{sect3.3}.
\end{enumerate}
\end{theo*}

We choose the hyperplane defined by $x_d=0$ to be $H$ in Problem~\ref{pro1}, and consider for any $n\geq m$ in $\mathbb N^*$ the so-called $(n,m)$-iterated contact locus $\mathscr{X}_{n,m,O}(f,x_d)$ of the pair $(f,x_d)$. It is a $\mu_n$-invariant $\mathbb C$-subvariety of $\mathscr X_{n,O}(f)$. Then we show in this article that the formal series
\[
Z_{f,x_d,O}^{\Delta}(T):=\sum_{n\geq m\geq 1}\bigl[\mathscr{X}_{n,m,O}(f,x_d)\bigr]\LL^{-(n+m)d}T^n
\]
is rational and it can be described via data of $\Gamma$. Here, $\Delta$ stands for $\{(n,m)\in (\mathbb R_{>0})^2\mid n\geq m\}$ and the sum runs over $\Delta\cap (\mathbb N^*)^2$. Put $\mathscr S_{f,x_d,O}^{\Delta}:=-\lim_{T\to \infty}Z_{f,x_d,O}^{\Delta}(T)$. Using the description of $\mathscr S_{f,x_d,O}^{\Delta}$ together with Theorem~\ref{thm4.1}, a solution to Problem~\ref{pro1} for the nondegeneracy in the sense of Kouchnirenko can be realized as in the following theorem.

\begin{theo*}[see Theorem~\ref{restricting}]
With $f$ as previous, the identity $\mathscr{S}_{f,O}=\mathscr{S}_{f|_H,O}+\mathscr S_{f,x_d,O}^{\Delta}$ holds in the monodromic Grothendieck ring of $\mathbb C$-varieties with $\hat\mu$-action. A similar result also holds for the motivic nearby cycles.
\end{theo*}

According to~\cite[Conjecture~1.5]{BFLN}, it is expected that the singular cohomology groups with compact support of the $\mathbb C$-points of the contact loci are nothing but the Floer cohomology groups of the powers of the monodromy of the singularity (cf.~\cite{Mc}). Here, we are interested in a smaller problem on the computation of cohomology groups of $\mathscr X_{n,O}(f)$ (the reader may compare this with~\cite[Theorem~1.1]{BFLN}).

\begin{enonce}{Problem}\label{pro2}
Let $f$ be a polynomial over $\mathbb C$ vanishing at the origin $O$. Compute the cohomology groups with compact support $H_c^m(\mathscr X_{n,O}(f),\mathbb C)$ for all $n\in \mathbb N^*$ and $m\in\mathbb N$.
\end{enonce}

We devote Section~\ref{Sec5-cloci} to study this problem for nondegenerate singularities in the sense of Kouchnirenko not only using sheaf cohomology with compact support but also the Borel--Moore homology $H_*^{\BM}$. Write $\mathscr X_{n,O}(f)=\bigsqcup_{(J,a)\in \widetilde{\mathcal P}_n}\mathscr{X}_{J,a}^{(n)}$ as in~\eqref{otherdecomp} with $\widetilde{\mathcal P}_n$ described in Lemma~\ref{indexPn}$\MK$\eqref{8ii}. Let $\eta: \widetilde{\mathcal P}_n\to \mathbb{Z}$ be the function defined by $\eta(J,a)=\dim_{\mathbb{C}}\mathscr{X}_{J,a}^{(n)}$.

\begin{theo*}[see Theorems~\ref{thm5.4}, \ref{thm5.6}]
For $f$ as in Problem~\ref{pro2} and nondegenerate in the sense of Kouchnirenko, for every $p, q\in \mathbb N$, there exist spectral sequences
\begin{gather*}
E_{p,q}^1:=\bigoplus_{\eta(J, a)=p}H^{\BM}_{p+q}(\mathscr{X}_{J,a}^{(n)})\Longrightarrow H^{\BM}_{p+q}(\mathscr X_{n,O}(f)),\\
E^{p,q}_1:=\bigoplus_{\eta(J,a)=p}H^{p+q}_c(\mathscr{X}_{J,a}^{(n)}, \mathcal{F})\Longrightarrow H^{p+q}_c(\mathscr X_{n,O}(f), \mathcal{F}),
\end{gather*}
for any sheaf of abelian groups $\mathcal{F}$ on $\mathscr X_{n,O}(f)$.
\end{theo*}

In particular, by applying the second spectral sequence with $ \mathcal{F}$ being a constant sheaf, we obtain a spectral sequence converging to the compact support cohomology groups of contact loci with complex coefficients whose first page is a direct sum of (singular) homology of the spaces defined by the vanishing of the functions $f_{\gamma}$ and $f_{\gamma}-1$ (see Corollary~\ref{corconstantsheaf}).


\section{Preliminaries}\label{sec2}

\subsection{Monodromic Grothendieck ring of varieties}\label{sect2.1}

Let $S$ be an algebraic $\mathbb C$-variety. Let $\Var_S$ be the category of $S$-varieties, with objects being morphisms of algebraic $\mathbb C$-varieties $X\to S$ and a morphism in $\Var_S$ from $X\to S$ to $Y\to S$ being a morphism of algebraic $\mathbb C$-varieties $X\to Y$ commuting with $X\to S$ and $Y\to S$. Denote by $\hat{\mu}$ the limit of the projective system $\mu_{nm}\to \mu_n$ given by $x\mapsto x^m$, with for any $n\geq 1$, $\mu_n=\Spec\, {\mathbb{C}[\xi]/(\xi^n-1)}$ the group scheme over $\mathbb C$ of $n$th roots of unity. Notice that any action of $\hat\mu$ on a variety $X$ in the present article is assumed to factorize through an action of $\mu_n$ for some $n\in \mathbb N^*$. An action on $X$ is good if every orbit is contained in an affine open subset of $X$. By definition, an action of $\hat\mu$ on an affine Zariski bundle $X\to B$ is affine if it is a lifting of a good action on $B$ and its restriction to all fibers is affine.

The Grothendick group $K_0^{\hat\mu}(\Var_S)$ is defined to be an abelian group generated by symbols $[X\to S]$, $X$ endowed with a good $\hat{\mu}$-action and $X\to S$ in $\Var_S$, such that:
\begin{enumerate}\romanenumi
\item $[X\to S]=[Y\to S]$ if $X$ and $Y$ are $\hat\mu$-equivariant $S$-isomorphic;
\item $[X\to S]=[Y\to S]+[X\setminus Y\to S]$ if $Y$ is a $\hat\mu$-invariant closed subvariety in $X$; and
\item $[X\times\mathbb{A}_{\mathbb C}^n,\sigma]=[X\times\mathbb{A}_{\mathbb C}^n,\sigma']$ if $\sigma$ and $\sigma'$ are liftings of the same $\hat{\mu}$-action on $X$ to $X\times\mathbb{A}_{\mathbb C}^n$.
\end{enumerate}
There is a natural ring structure on $K_0^{\hat\mu}(\Var_S)$ in which the product is induced by the fiber product over $S$. The unit $1_S$ for the product is the class of the identity morphism $S\to S$ with $S$ endowed with trivial $\hat\mu$-action. Denote by $\LL$ (or $\LL_S$) the class of the trivial line bundle $S\times\mathbb{A}^1\to S$, and define the localized ring $\mathscr{M}_S^{\hat{\mu}}$ to be $K_0^{\hat\mu}(\Var_S)[\LL^{-1}]$.

Let $f:S\to S'$ be a morphism of algebraic $\mathbb C$-varieties. Then we have two important morphisms associated to $f$, which are the ring homomorphism $f^*: \mathscr{M}_{S'}^{\hat{\mu}}\to \mathscr{M}_S^{\hat{\mu}}$ induced from the fiber product (the pullback morphism) and the $\mathscr M_{\mathbb C}$-linear homomorphism $f_!: \mathscr{M}_S^{\hat{\mu}}\to \mathscr{M}_{S'}^{\hat{\mu}}$ defined by the composition with $f$ (the push-forward morphism).

\subsection{Rational series and limit}\label{2.3}

Let $\mathscr A$ be either $\mathbb{Z}[\LL,\LL^{-1}]$ or $\mathscr{M}_S^{\hat\mu}$ as a ring. Let $\mathscr A\llbracket T\rrbracket _{\sr}$ be the $\mathscr A$-submodule of $\mathscr A\llbracket T\rrbracket $ generated by $1$ and by finite products of elements of the form $\frac{\LL^aT^b}{1-\LL^aT^b}$ with $(a,b)$ in $\mathbb{Z}\times\mathbb{N}^*$. Each element of $\mathscr A\llbracket T\rrbracket _{\sr}$ is called a \emph{rational series}. By~\cite{DL1}, there is a unique $\mathscr A$-linear morphism $\lim_{T\rightarrow\infty}: \mathscr A\llbracket T\rrbracket _{\sr}\rightarrow \mathscr A$ which sends $\frac{\LL^aT^b}{1-\LL^aT^b}$ to $-1$.

For $J$ contained in $[d]$, we denote by $(\mathbb{R}_{\geq 0})^J$ the set of $(a_j)_{j\in J}$ with $a_j$ in $\mathbb{R}_{\geq 0}$ for all $j\in J$, and by $(\mathbb{R}_{>0})^J$ the subset of $(\mathbb{R}_{\geq 0})^J$ consisting of $(a_j)_{j\in J}$ with $a_j>0$ for all $j\in J$. Similarly, one can define the sets $(\mathbb{Z}_{\geq 0})^J$, $(\mathbb{Z}_{>0})^J$ and $(\mathbb{N}^*)^J$. Let $\sigma$ be a rational polyhedral convex cone in $(\mathbb{R}_{>0})^J$ and let $\overline{\sigma}$ denote its closure in $(\mathbb{R}_{\geq 0})^J$ with $J$ a finite set. Let $\ell$ and $\ell'$ be two integer linear forms on $\mathbb{Z}^J$ positive on $\overline{\sigma}\setminus\{(0,\dots,0)\}$. Then the series
\[
S_{\sigma,\ell,\ell'}(T):=\sum_{a\in\sigma\cap(\mathbb{N}^*)^J}\mathbb{L}^{-\ell'(a)}T^{\ell(a)}
\]
is in $\mathbb{Z}[\mathbb{L},\mathbb{L}^{-1}]\llbracket T\rrbracket _{\sr}$ and $\lim_{T\rightarrow\infty}S_{\sigma,\ell,\ell'}(T)=\chi(\sigma)$, the Euler characteristic with compact supports of $\sigma$. If $\sigma$ is relatively open, then $\lim_{T\rightarrow\infty}S_{\sigma,\ell,\ell'}(T)=(-1)^{\dim(\sigma)}$ (see~\cite[Lemma~2.1.5]{G}). We have the following technique lemma.

\begin{lemm}\label{lem21}
Let $\sigma$ be a relatively open rational polyhedral convex cone in $(\mathbb R_{>0})^I$. Let $K$ and $L$ be disjoint nonempty subsets of $I$. Consider half spaces $H_j$ (with $j\in L$) in $\mathbb R^I$ defined by
\[
x_j\leq \sum_{i\in K}\alpha_ix_i,
\]
where $\alpha_i\geq 0$ for all $i\in K$, such that for any disjoint subsets $L_1, L_2$ of $L$ and any $j\in L\setminus (L_1\cup L_2)$, the set
\[
\sigma\cap \bigcap_{s\in L_1}\left\{(x_i)_{i\in I}\in \mathbb R^I \,\middle|\, x_{s}<\sum_{i\in K}\alpha_ix_i\right\}\cap \bigcap_{t\in L_2}\left\{(x_i)_{i\in I}\in \mathbb R^I \,\middle|\, x_{t}=\sum_{i\in K}\alpha_ix_i\right\}
\]
either is empty or has nonempty intersection with $\mathbb R^I\setminus H_j$.

Then the Euler characteristic with compact supports of the set
\[
\sigma_L:=\sigma \cap \bigcap_{j\in L}H_j
\]
is equal to zero.
\end{lemm}

\begin{proof}
We prove this lemma by induction on $|L|$. For $L=\{j\}$ a one-point set, we have
\[
\sigma= \sigma_L \sqcup \left\{(x_i)_{i\in I}\in \sigma \,\middle|\, x_j> \sum_{i\in K}\alpha_ix_i\right\}.
\]
Since the second term on the right hand side and $\sigma$ have the same Euler characteristic with compact supports $(-1)^{|I|}$, we get $\chi(\sigma_L)=0$. For the case $|L|>1$, let $j_0\in L$ and $L':=L\setminus \{j_0\}$. Then $\sigma_L$ is the disjoint union of the following two sets
\[
\sigma_{L'}\cap \left\{(x_i)_{i\in I}\in \mathbb R^I \,\middle|\, x_{j_0}<\sum_{i\in K}\alpha_ix_i\right\}
\]
and
\[
\sigma_{L'}\cap \left\{(x_i)_{i\in I}\in \mathbb R^I \,\middle|\, x_{j_0}=\sum_{i\in K}\alpha_ix_i\right\}.
\]
By induction, the lemma holds true for $L'$, thus the Euler characteristic with compact supports of these two sets is zero.
\end{proof}


\subsection{Motivic nearby cycles of regular functions}\label{section2.3}
For any $\mathbb C$-variety $X$, let $\mathscr{L}_n(X)$ be the space of $n$-jets on $X$, and $\mathscr{L}(X)$ the arc space on $X$, which is the limit of the projective system of spaces $\mathscr{L}_n(X)$ and canonical morphisms $\mathscr{L}_m(X)\to \mathscr{L}_n(X)$ for $m\geq n$. The group $\hat\mu$ acts on $\mathscr{L}_n(X)$ via $\mu_n$ in such a natural way that $\xi\cdot\varphi(t)=\varphi(\xi t)$ for $\xi\in \mu_n$.

From now on, we assume that the $\mathbb C$-variety $X$ is smooth and of pure dimension $d$. Consider a regular function $f:X\to \mathbb{A}_{\mathbb C}^1$, with the zero locus $X_0$. For $n\geq 1$ one defines the \emph{$n$-iterated contact locus} of $f$ as follows
\[
\mathscr{X}_n(f)=\left\{\varphi\in\mathscr{L}_n(X)\,\middle|\, f(\varphi)=t^n\bmod t^{n+1}\right\}.
\]
Clearly, this variety is invariant by the $\hat\mu$-action on $\mathscr{L}_n(X)$ and admits a morphism to $X_0$ given by $\varphi(t)\mapsto \varphi(0)$, which defines an element $[\mathscr{X}_n(f)]:=[\mathscr{X}_n(f)\to X_0]$ in $\mathscr{M}_{X_0}^{\hat\mu}$. We consider Denef--Loeser's motivic zeta function $Z_f(T)=\sum_{n\geq 1}[\mathscr{X}_n(f)]\LL^{-nd}T^n.$ They prove in~\cite{DL1} that $Z_f(T)$ is in $\mathscr{M}_{X_0}^{\hat\mu}\llbracket T\rrbracket _{\sr}$, and call the limit $\mathscr S_f:=-\lim_{T\to\infty}Z_f(T)$ in $\mathscr{M}_{X_0}^{\hat\mu}$ the \emph{motivic nearby cycles} of $f$. If $x$ is a closed point of $X_0$, the $\mathbb C$-variety
\[
\mathscr{X}_{n,x}(f)=\left\{\varphi\in\mathscr{L}_n(X)\,\middle|\, f(\varphi)=t^n\bmod t^{n+1}, \varphi(0)=x\right\},
\]
is also invariant by the $\hat\mu$-action on $\mathscr L_n(X)$, called the \emph{$n$-iterated contact locus} of $f$ at $x$. It is also proved that the zeta function $Z_{f,x}(T)=\sum_{n\geq 1}[\mathscr{X}_{n,x}(f)]\LL^{-nd}T^n$ is in $\mathscr{M}_{\mathbb C}^{\hat\mu}\llbracket T\rrbracket _{\sr}$. The limit $\mathscr S_{f,x}=-\lim_{T\to\infty}Z_{f,x}(T)$ is called the \emph{motivic Milnor fiber} of $f$ at $x$. Obviously, if $\iota$ is the inclusion of $\{x\}$ in $X_0$, then $\mathscr S_{f,x}=\iota^*\mathscr S_f$ in $\mathscr{M}_{\mathbb C}^{\hat\mu}$.

We now modify slightly the motivic zeta functions of several functions in~\cite{G} and~\cite{GLM2}. For a pair of regular functions $(f,g)$ on $X$, we denote by $X_0:=X_0(f,g)$ their common zero locus. For $n\geq m$ in $\mathbb N^*$, we define
\[
\mathscr X_{n,m}(f,g):=\left\{\varphi\in\mathscr{L}_n(X)\,\middle|\, f(\varphi)= t^n\bmod t^{n+1}, \ord_tg(\varphi)=m\right\}.
\]
We can check that $\mathscr{X}_{n,m}(f,g)$ is invariant under the natural $\mu_n$-action on $\mathscr{L}_n(X)$, and that there is an obvious morphism of $\mathbb C$-varieties $\mathscr{X}_{n,m}(f,g)\to X_0$; from which we obtain the class $[\mathscr{X}_{n,m}(f,g)]$ of that morphism in $\mathscr{M}_{X_0}^{\hat\mu}$. Consider the series
\[
Z_{f,g}^{\Delta}(T):=\sum_{n\geq m\geq 1}\bigl[\mathscr X_{n,m}(f,g)\bigr]\LL^{-nd}T^n
\]
in $\mathscr M_{X_0}^{\hat\mu}\llbracket T\rrbracket $. For any closed point $x\in X_0$, we can define $Z_{f,g,x}^{\Delta}(T)$ in $\mathscr M_{\mathbb C}^{\hat\mu}\llbracket T\rrbracket $ as above with $\mathscr{X}_{n,m}(f,g)$ replaced by its $\mu_n$-invariant subvariety $\mathscr{X}_{n,m,x}(f,g):=\{\varphi\in \mathscr{X}_{n,m}(f,g)\mid \varphi(0)=x\}$. The rationality of the series $Z_{f,g}^{\Delta}(T)$ and $Z_{f,g,x}^{\Delta}(T)$ are stated in~\cite[Th\'eor\`eme~4.1.2]{G} and~\cite[Section~2.9]{GLM2}, up to the isomorphism of rings $\mathscr{M}_{X_0}^{\hat{\mu}}\cong\mathscr{M}_{X_0\times\mathbb{G}_m}^{\mathbb{G}_m}$ (see~\cite[Proposition~2.6]{GLM1}), where Guibert--Loeser--Merle's result is done in the framework $\mathscr{M}_{X_0\times\mathbb{G}_m}^{\mathbb{G}_m}$. Put $\mathscr S_{f,g}^{\Delta}:=-\lim_{T\to \infty}Z_{f,g}^{\Delta}(T)$ and $\mathscr S_{f,g,x}^{\Delta}:=-\lim_{T\to \infty}Z_{f,g,x}^{\Delta}(T)$.


\section{Motivic nearby cycles of a nondegenerate polynomial and applications}\label{sec3}

\subsection{Newton polyhedron of a polynomial}\label{nondegenerate}

Recall that $[d]$ stands for $\{1,\dots,d\}$, $d\in \mathbb N^*$. Let $x=(x_1,\dots,x_d)$ be a set of $d$ variables, and let $f(x)=\sum_{\alpha\in\mathbb{N}^d}c_{\alpha}x^{\alpha}$ be in $\mathbb C[x]$ with $f(O)=0$, with $O$ the origin of $\mathbb C^d$. Let $\Gamma$ be the Newton polyhedron of $f$, i.e., the convex hull of the set $\bigcup_{c_{\alpha}\not=0}(\alpha+(\mathbb{R}_{\geq 0})^d)$ in $(\mathbb{R}_{\geq 0})^d$. For every face $\gamma$ of $\Gamma$ (not necessarily compact, the case $\gamma=\Gamma$ included), define by $f_{\gamma}(x)=\sum_{\alpha\in\gamma}c_{\alpha}x^{\alpha}$ the \emph{face function} of $f$ with respect to $\gamma$.

Note that for every face $\gamma$ of $\Gamma$ (including $\Gamma$ itself), there exists a unique set $J_{\gamma}\subseteq [d]$ such that $\gamma$ is contained in the hyperplanes $x_j=0$ for all $j\not\in J_{\gamma}$ and not contained in other coordinate hyperplanes.

\begin{defi}
The polynomial $f$ is called \emph{nondegenerate on a face $\gamma$} of $\Gamma$ if the hypersurface $f_{\gamma}^{-1}(0)$ has no singular point in $\mathbb{G}_{m,\mathbb C}^{J_{\gamma}}$. We say that $f$ is \emph{nondegenerate in the sense of Kouchnirenko} if it is nondegenerate on every compact face $\gamma$. If $f$ is nondegenerate on every face of $\Gamma$ (including non-compact faces, and $\Gamma$ itself), we say that $f$ is \emph{nondegenerate in the sense of Newton polyhedron} or simply \emph{Newton nondegenerate}.
\end{defi}

Consider the function $\ell=\ell_{\Gamma}: (\mathbb R_{\geq 0})^d \to \mathbb R$ which sends $a$ in $(\mathbb R_{\geq 0})^d$ to $\min_{b\in\Gamma}\langle a,b\rangle$, where $\langle {\,\cdot\,,\cdot\,}\rangle$ is the standard inner product in $\mathbb R^d$. For $a$ in $(\mathbb{R}_{\geq 0})^d$, we denote by $\gamma_a$ the maximal face of $\Gamma$ to which the restriction of the function $\langle a,\cdot\,\rangle$ gets its minimum. Note that $\gamma_a$ is a compact face if and only if $a$ is in $(\mathbb{R}_{>0})^d$ (cf.~\cite[Property 2.3]{DH}). This comes from the fact that $\gamma_a=\{b\in\Gamma \mid \langle a,b\rangle=\ell(a)\}$. Moreover, $\gamma_a=\Gamma$ when $a=(0,\dots,0)$ in $\mathbb R^d$, and $\gamma_a$ is a proper face of $\Gamma$ otherwise. For every proper face $\gamma$ of $\Gamma$, we define
\[
\sigma_{\gamma}:=\sigma_{[d],\gamma}:=\bigl\{a\in(\mathbb{R}_{\geq 0})^d\,\big|\, \gamma=\gamma_a\bigr\}.
\]
It is clear that $\sigma_{\gamma}$ is a rational polyhedral convex cone of dimension $d-\dim(\gamma)$.

For any $J\subseteq [d]$, denote by $f^J$ the polynomial in $\mathbb C[(x_j)_{j\in J}]$ obtained from $f(x)$ substituting $x_i$ by $0$ for all $i\in [d]\setminus J$. If $f$ is nondegenerate in the sense of Kouchnirenko (resp. Newton nondegenerate) then $f^J$ is also nondegenerate in the sense of Kouchnirenko (resp. Newton polyhedron).

Let $\ell_J$ stand for $\ell_{\Gamma(f^J)}$. For $a\in (\mathbb{R}_{\geq 0})^J$, we define the face $\gamma^J_a$ similarly as above, i.e.
\[
\gamma_a^J:=\bigl\{b\in \Gamma(f^J)\big| \langle a,b\rangle=\ell_J(a)\bigr\}.
\]
If $\gamma$ is a face of the Newton polyhedron $\Gamma(f^J)$, denote by $\sigma_{J,\gamma}$ the cone $\{a\in (\mathbb{R}_{\geq 0})^J\mid \gamma=\gamma_a^J\}$ and by $\mathring{\sigma}_{J,\gamma}$ the relative interior of $\sigma_{J,\gamma}$, both of which have dimension $|J|-\dim(\gamma)$. The following lemma is trivial to prove.

\begin{lemm}\label{partition}
There exists a canonical partition of $(\mathbb R_{\geq 0})^J$ into rational polyhedral convex cones $\mathring{\sigma}_{J,\gamma}$ with $\gamma$ being all faces of $\Gamma(f^J)$.
\end{lemm}

A face $\gamma$ of $\Gamma$ is called a \emph{coordinate face} if $\gamma=\Gamma(f^J)$ for some $J\subseteq [d]$. We have the following description for the coordinate faces.

\begin{lemm}\label{coordinateface}
A face $\gamma$ of\/ $\Gamma$ is a coordinate face if and only if for all $J\supseteq J_{\gamma}$, the restriction of $\ell_{J}$ to $\mathring{\sigma}_{J,\gamma}$ is the zero function.
\end{lemm}

\begin{proof}
For each $j\in J$, we denote by $e^j$ the vector $(0, \ldots, 0, 1, 0, \ldots, 0)\in \mathbb{R}^J$ with $1$ in the $j$-th coordinate. Assume that $\gamma=\Gamma(f^{J_0})$ for some $J_0\subseteq [d]$. Then, for any $i\in J_0$, $t \geq 0$ and for any $b\in \gamma$, the point $b+te^i$ is also in $\gamma$. For $J\supseteq J_0= J_{\gamma}$ and $a = (a_j)_{j\in J}\in \mathring{\sigma}_{J,\gamma}$, we have $\ell_J(a)= \langle a, b+te^i \rangle $ for every $i\in J_0$ and $t\geq 0$. As a consequence, we get $a_j=0$ for all $j\in J_0$. Hence $\ell_J(a)= \langle a, b \rangle=0 $ for any $b\in \gamma=\Gamma(f^{J_0})$.

Now, we assume that restriction of $\ell_{J}$ to $\mathring{\sigma}_{J,\gamma}$ is the zero function for some $J\supseteq J_{\gamma}$. Take $a = (a_j)_{j\in J}\in \mathring{\sigma}_{J,\gamma}$, we have $\langle a, b \rangle=\ell_J(a)= 0 $ for any $b\in \gamma$. Since $\gamma$ is not contained in any hyperplane $x_k=0$ for any $k\in J_{\gamma}$, we have $a_j=0$ for all $j\in J_{\gamma}$. This together with the description of $\gamma$, namely,
\[
\gamma = \bigl\{b \in \Gamma(f^{J})\,\big|\, \langle a, b\rangle=0\bigr\},
\]
implies that $\Gamma(f^{J_{\gamma}})\subseteq \gamma$. Therefore $\gamma = \Gamma(f^{J_{\gamma}})$.
\end{proof}

\begin{nota}
In the rest of this article, let $F$ (resp. $K$) denote the set of all the faces (resp. the compact faces) of $\Gamma$, and let $\widetilde{F}$ denote the set of all the coordinate faces of $\Gamma$.
\end{nota}

\subsection{Contact loci}\label{sect3.2}
Let $(x_1,\dots,x_d)$ be the standard coordinates of $\mathbb A^d_{\mathbb C}$, and let $f(x_1,\dots,x_d)$ be as above. For $n\in \mathbb N^*$, $k\in \mathbb N$ and $J\subseteq [d]$, denote by $\Delta_J^{(n,k)}$ (resp. $\widetilde\Delta_J^{(n,k)}$) the set of $a\in \{0,\dots,n\}^J$ (resp. $a\in [n]^J$) such that $\ell_J(a)+k=n$. Clearly, $\widetilde\Delta_J^{(n,k)}\subseteq \Delta_J^{(n,k)}$.

For $a\in \Delta_J^{(n,k)}$, put
\[
\mathscr{X}_{J,a}^{(n)}:=\left\{\varphi\in\mathscr{X}_n(f)\,\middle|\, \ord_tx_j(\varphi)=a_j \ \forall j\in J,\ x_i(\varphi)\equiv 0\ \forall i\not\in J\right\}.
\]
This subvariety of $\mathscr{X}_n(f)$ is invariant by the $\mu_n$-action given by $\xi\cdot\varphi(t)=\varphi(\xi t)$, and it defines an element $\bigl[\mathscr{X}_{J,a}^{(n)}\bigr]:=\bigl[\mathscr{X}_{J,a}^{(n)}\to X_0\bigr]$ in $K_0^{\hat\mu}(\Var_{X_0})$, where the structure map is given by $\varphi\mapsto \varphi(0)$. Let $\mathcal{P}_n$ and $\widetilde{\mathcal P}_n$ be the index sets consisting of all such pairs $(J,a)$ such that
\begin{align}\label{decompXn}
\mathscr X_{n}(f)=\bigsqcup_{(J,a)\in \mathcal{P}_n}\mathscr{X}_{J,a}^{(n)}
\end{align}
and
\begin{align}\label{otherdecomp}
\mathscr X_{n,O}(f)=\bigsqcup_{(J,a)\in \widetilde{\mathcal P}_n}\mathscr{X}_{J,a}^{(n)}.
\end{align}

\begin{lemm}\label{indexPn}\ 
\begin{enumerate}\romanenumi
\item \label{8i} $\mathcal P_n$ is the set of all the pairs $(J,a)$ such that $J\supseteq J_{\gamma}$, $a\in \bigsqcup_{k\in \mathbb N}\big(\mathring{\sigma}_{J,\gamma}\cap \Delta_J^{(n,k)}\big)$ and $\gamma\in F$.
\item \label{8ii} $\widetilde{\mathcal P}_n$ is the set of all the pairs $(J,a)$ such that $J\supseteq J_{\gamma}$, $a\in \bigsqcup_{k\in \mathbb N}\big(\mathring{\sigma}_{J,\gamma}\cap \widetilde\Delta_J^{(n,k)}\big)$ and $\gamma\in K$.
\end{enumerate}
\end{lemm}

\begin{proof}
For any $\varphi$ in $\mathscr X_{n}(f)$, there exists a unique subset $J$ of $\{1,\dots,d\}$ such that $x_i(\varphi)\equiv 0$ for all $i\not\in J$ and that $x_j(\varphi)\not\equiv 0$ for all $j\in J$. Put $a:=(\ord_t x_j(\varphi))_{j\in J}\in \{0,\dots,n\}^J$, and put $\gamma:=\gamma_a^J$. Then we have $J_{\gamma}\subseteq J$ and
\[
f(\varphi)=f_{\gamma}(\widetilde\varphi(0))t^{\ell_J(a)}+\text{higher terms},
\]
where $\widetilde\varphi:=(t^{-a_j}x_j(\varphi))_{j\in J}$, thus $\ell_J(a)\leq n$ and $\varphi\in \mathscr X_{J,a}^{(n)}$. The proof for~\eqref{8i} is completed by using Lemma~\ref{partition}. Similar arguments work for~\eqref{8ii}.
\end{proof}

For $\gamma\in \widetilde{F}$, if $a\in \mathring{\sigma}_{J,\gamma}\cap \Delta_J^{(n,k)}$ and $k\not=n$, then $\mathscr{X}_{J,a}^{(n)}=\emptyset$.


For every face $\gamma\in F$ of $\Gamma(f^J)$, let us consider the $\mathbb C$-varieties
\[
X_{J,\gamma}(1):=\bigl\{x\in \mathbb G_{m,\mathbb C}^J \,\big|\, f_{\gamma}(x)=1\bigr\},\quad X_{J,\gamma}(0):=\bigl\{x\in \mathbb G_{m,\mathbb C}^J \,\big|\, f_{\gamma}(x)=0\bigr\}.
\]
When $J=J_{\gamma}$ we write simply $X_{\gamma}(\varepsilon)$ instead of $X_{J_{\gamma},\gamma}(\varepsilon)$, for $\varepsilon=0, 1$. We always consider the trivial action of $\hat\mu$ on the variety $X_{J,\gamma}(0)$. Let $a$ be in $\mathring{\sigma}_{J,\gamma}$. Then the variety $X_{J,\gamma}(1)$ admits a natural $\mu_{\ell_J(a)}$-action as follows
\begin{align}\label{action}
e^{2\pi ir/\ell_J(a)}\cdot (x_j)_{j\in J}:=\big(e^{2\pi ira_j/\ell_J(a)}x_j\big)_{j\in J},
\end{align}
for $r\in [\ell_J(a)]$. Note that the class $[X_{J,\gamma}(1)]$ in $\mathscr M_{\mathbb C}^{\mu_n}$ does not depend on $a$ provided $a$ is in $\mathring{\sigma}_{J,\gamma}$ and $\ell_J(a)=n$, which follows from the construction of the Grothendieck ring (see~\cite[Proposition~3.13]{Raibaut}).

The result and proof ideas of the following theorem are well known due to~\cite{G, AB-CN-L-MH, GLM2}. In the present article, we are going to contribute a detailed explanation for every step of proof. Denote $|a|:=\sum_{j\in J}a_j$ for $a=(a_j)_{j\in J}\in \mathbb R^J$.

\begin{theo}[cf.~\cite{G, AB-CN-L-MH, GLM2}]\label{mainthm1}
Let $f\in \mathbb{C}[x_1,\dots,x_d]$ such that $f(O)=0$. Assume that $f$ is nondegenerate on a face $\gamma\in F$. Let $J\subseteq [d]$ containing $J_{\gamma}$. If $a\in \mathring{\sigma}_{J,\gamma}\cap \Delta_J^{(n,0)}$ and $\mathscr{X}_{J,a}^{(n)}$ is nonempty, then there is a naturally $\mu_n$-equivariant isomorphism of $\mathbb{C}$-varieties
\[
\tau: \mathscr{X}_{J,a}^{(n)}\to X_{J,\gamma}(1) \times_{\mathbb C}\mathbb A_{\mathbb C}^{|J|\ell_J(a)-|a|}.
\]
If $k\in \mathbb N^*$, $a\in \mathring{\sigma}_{J,\gamma}\cap \Delta_J^{(n,k)}$, and if $\mathscr{X}_{J,a}^{(n)}$ is nonempty, there is a Zariski locally trivial fibration
\[
\pi: \mathscr{X}_{J,a}^{(n)}\to X_{J,\gamma}(0)
\]
with fiber $\mathbb A_{\mathbb C}^{|J|(\ell_J(a)+k)-|a|-k}$.
\end{theo}

\begin{proof}
It suffices to prove the theorem for $J=[d]$. Let $a=(a_1,\dots,a_d)$ be in $\mathring{\sigma}_{\gamma}\cap \Delta_{[d]}^{(n,0)}$, hence $n=\ell(a)$ and $\gamma=\gamma_a$. Every element $\varphi$ in $\mathscr{X}_{[d],a}^{(n)}$ has the form
\[
\left(\sum_{j=a_1}^{\ell(a)}b_{1j}t^j,\dots, \sum_{j=a_d}^{\ell(a)}b_{dj}t^j\right)
\]
with $b_{ia_i}\not=0$ for $1\leq i\leq d$. The coefficient of $t^{\ell(a)}$ in $f(\varphi(t))$ is nothing but $f_{\gamma_a}(b_{1a_1},\dots,b_{da_d})$, thus $(b_{1a_1},\dots,b_{da_d})$ is in $X_{[d],\gamma_a}(1)$. We deduce that $\mathscr{X}_{[d],a}^{(\ell(a))}$ is $\mu_{\ell(a)}$-equivariant isomorphic to $X_{[d],\gamma_a}(1)\times_{\mathbb C}\mathbb{A}_{\mathbb C}^{d\ell(a)-|a|}$ (where $\mu_{\ell(a)}$ acts trivially on $\mathbb{A}_{\mathbb C}^{d\ell(a)-|a|}$) via the map
\[
\tau: \varphi(t)\mapsto \left((b_{ia_i})_{1\leq i\leq d},(b_{ij})_{1\leq i\leq d,a_i<j\leq \ell(a)}\right).
\]
Indeed, for every $\xi$ in $\mu_{\ell(a)}$, the element $\varphi(\xi t)$ is sent to
\[
\left((\xi^{a_i}b_{ia_i})_{1\leq i\leq d},(b_{ij})_{1\leq i\leq d,a_i<j\leq \ell(a)}\right)=\xi\cdot\left((b_{ia_i})_{1\leq i\leq d},(b_{ij})_{1\leq i\leq d,a_i<j\leq \ell(a)}\right).
\]
Thus $\tau$ is a $\mu_{\ell(a)}$-equivariant isomorphism.

Now we prove the second statement. Let $a$ be in $\mathring{\sigma}_{\gamma}\cap \Delta_{[d]}^{(n,k)}$ for $k\in \mathbb N^*$, hence $n=\ell(a)+k$ and $\gamma=\gamma_a$. For $\varphi$ in $\mathscr X_{[d],a}^{(n)}$, putting
\begin{align}\label{tildephi}
\widetilde{\varphi}:=\big(t^{-a_1}x_1(\varphi),\dots,t^{-a_d}x_d(\varphi)\big),
\end{align}
we get
\begin{equation}\label{onestar}
f(\varphi)=t^{\ell(a)}f_{\gamma_a}(\widetilde{\varphi})+\sum_{k\geq 1}t^{\ell(a)+k}\sum_{\langle \alpha,a \rangle=\ell(a)+k}c_{\alpha}\widetilde{\varphi}^{\alpha}.
\end{equation}
Defining
\[
\widetilde f(\widetilde{\varphi},t):=f_{\gamma_a}(\widetilde{\varphi})+\sum_{k\geq 1}t^k\sum_{\langle \alpha,a \rangle=\ell(a)+k}c_{\alpha}\widetilde{\varphi}^{\alpha},
\]
we obtain a function
\[
\widetilde f: \mathscr L_{\ell(a)+k+1-a_1}(\mathbb A_{\mathbb C}^1)\times_{\mathbb C}\cdots \times_{\mathbb C}\mathscr L_{\ell(a)+k+1-a_d}(\mathbb A_{\mathbb C}^1)\times_{\mathbb C} \mathbb A_{\mathbb C}^1 \to \mathbb A_{\mathbb C}^1
\]
given by
\[
\widetilde f(\widetilde{\varphi},t_0):=\widetilde f(\widetilde{\varphi}(t_0),t_0).
\]
It thus follows from~\eqref{onestar} that $\varphi$ is in $\mathscr X_{[d],a}^{(\ell(a)+k)}$ if and only if $\widetilde f(\widetilde{\varphi},t)=t^k\mod t^{k+1}$. Putting $\widetilde{\varphi}_i(t)=\sum_{j=0}^{\ell(a)-a_i+k}b_{ij}t^j$ for $1\leq i\leq d$, the latter means that
\[
\begin{cases}
f_{\gamma_a}(b_{10},\dots,b_{d0})=0 & \quad \text{with} \ b_{i0}\not=0 \ \text{for}\ 1\leq i\leq d,\\
q_j(b_{1j},\dots,b_{dj})+p_j((b_{i'j'})_{i',j'})=0 & \quad \text{for}\ 1\leq j\leq k-1,\\
q_k(b_{1k},\dots,b_{dk})+p_k((b_{i'j'})_{i',j'})=1,
\end{cases}
\]
where $p_j$, for $1\leq j\leq k$, are polynomials in variables $b_{i'j'}$ with $i^{'}\leq d$ and $j'<j$, and
\[
q_j(b_{1j},\dots,b_{dj})=\sum_{i=1}^{d}\frac{\partial f_{\gamma_a}}{\partial x_i}(b_{10},\dots,b_{d0})b_{ij}.
\]

We consider the morphism
\[
\pi: \mathscr X_{[d],a}^{(\ell(a)+k)}\to X_{[d],\gamma_a}(0)
\]
which sends the $\varphi$ described previously to $(b_{10},\dots,b_{d0})$. Since $\hat\mu$ acts trivially on $X_{[d],\gamma_a}(0)$, we only need to prove that $\pi$ is a locally trivial fibration with fiber $\mathbb A_{\mathbb C}^{d(\ell(a)+k)-|a|-k}$. For every $1\leq i\leq d$, we put
\begin{align}\label{trivialization}
U_i:=\left\{(x_1,\dots,x_d)\in X_{[d],\gamma_a}(0) \,\middle|\, \frac{\partial f_{\gamma_a}}{\partial x_i}(x_1,\dots,x_d)\not=0\right\}.
\end{align}
The nondegeneracy of $f$ on the face $\gamma=\gamma_a$ gives us an open covering $\{U_1,\dots,U_{d}\}$ of $X_{[d],\gamma}(0)$. We construct trivializations of $\pi$ as follows
\[
\xymatrix{
\pi^{-1}(U_i) \ar[rr]^{\Phi_{U_i}}\ar@{->}[dr]_{\pi}&& U_i\times_{\mathbb{C}} \mathbb{A}_{\mathbb{C}}^{e}\ar@{->}[dl]^{\pr_1}\\
&U_i& }
\]
where $e=\sum_{l=1}^d(\ell(a)-a_l+k)-k$ and we identify $\mathbb{A}_{\mathbb{C}}^{e}$ with the subvariety of $\mathbb{A}_{\mathbb{C}}^{\sum_{l=1}^d(\ell(a)-a_l+k)}$ defined by the equations $\widetilde b_{ij}=0$ for $1\leq j\leq k-1$ and $\widetilde b_{ik}=1$ in the coordinate system $(\widetilde b_{lj})$, and for $\varphi$ as previous,
\[
\Phi_{U_i}(\varphi)=\left((\widetilde b_{10},\dots,\widetilde b_{d0}),(\widetilde b_{lj})_{1\leq l\leq d, 1\leq j\leq \ell(a)-a_l+k}\right),
\]
with $\widetilde b_{ij}=0$ if $1\leq j\leq k-1$, $\widetilde b_{ik}=1$, and $\widetilde b_{lj}=b_{lj}$ otherwise. Furthermore, the inverse map $\Phi_{U_i}^{-1}$ of $\Phi_{U_i}$ is also a regular morphism given explicitly as follows
\[
\Phi_{U_i}^{-1}(\widetilde b_{lj})=\left(\sum_{j=0}^{\ell(a)-a_1+k}b_{1j}t^{j+a_1},\dots,\sum_{j=0}^{\ell(a)-a_d+k}b_{dj}t^{j+a_d}\right),
\]
where $b_{lj}=\widetilde b_{lj}$ for either that $l\not=i$ or that $l=i$ and $k<j\leq \ell(a)-a_l+k$, and
\[
b_{ij}=\frac{-p_j((b_{lj'})_{l\leq d, j'<j})-\sum_{l\leq d, l\not=i}(\partial f_{\gamma_a}/\partial x_l)(\widetilde b_{10},\dots,\widetilde b_{d0})\widetilde b_{lj}}{(\partial f_{\gamma_a}/\partial x_i)(\widetilde b_{10},\dots,\widetilde b_{d0})},
\]
for $1\leq j\leq k-1$, and
\[
b_{ik}=\frac{1-p_k((b_{lj'})_{l\leq d, j'<k})-\sum_{l\leq d, l\not=i}(\partial f_{\gamma_a}/\partial x_l)(\widetilde b_{10},\dots,\widetilde b_{d0})\widetilde b_{lk}}{(\partial f_{\gamma_a}/\partial x_i)(\widetilde b_{10},\dots,\widetilde b_{d0})}.
\]
This proves that $\pi$ is a (Zariski) locally trivial fibration with fiber $\mathbb A_{\mathbb C}^e$.
\end{proof}



\subsection{Motivic nearby cycles}\label{sect3.3}

For every $\gamma=\gamma_a^J\in F$ with $a\in \mathring{\sigma}_{J,\gamma}\cap \Delta_J^{(n,0)}$, we consider the morphism $\Phi_a: X_{J,\gamma}(1)\to X_0$ which sends $(x_i)_{i\in J}$ to $(\hat x_1,\dots,\hat x_d)$, where $\hat x_i=0$ if either $i\in [d]\setminus J$ or $a_i\geq 1$, and $\hat x_i=x_i$ if $a_i=0$. With this morphism it follows from Theorem~\ref{mainthm1} the below commutative diagram
\begin{equation}\label{eq3.5}
\begin{aligned}
\xymatrix{\mathscr X_{J,a}^{(n)}\qquad \ar[rr]^{\cong\qquad\quad}\ar@{->}[dr]_{\varphi(t)\mapsto \varphi(0)}&& \quad X_{J,\gamma}(1)\times_{\mathbb{C}} \mathbb{A}_{\mathbb{C}}^{|J|\ell_J(a)-|a|}\ar@{->}[dl]^{\Phi_a\circ \pr_1}\\
&X_0& }
\end{aligned}
\end{equation}

\begin{lemm}
If $a, b\in \mathring{\sigma}_{J,\gamma}\cap \Delta_J^{(n,0)}$, then $[\Phi_a: X_{J,\gamma}(1)\to X_0]=[\Phi_b: X_{J,\gamma}(1)\to X_0]$ in $\mathscr M_{X_0}^{\mu_n}$.
\end{lemm}

\begin{proof}
Suggested from~\cite[Section~3.4.3]{Raibaut}, we stratify $(\mathbb R_{\geq 0})^J$ into the cones
\[
C_{\delta}:=\left\{(k_i)_i\in (\mathbb R_{\geq 0})^J \,\middle|\, k_i>0 \ \text{if} \ \delta_i=1, k_i=0 \ \text{if} \ \delta_i=0\right\}
\]
with $\delta\in \{0,1\}^J$. We have a stratification of $\sigma_{J,\gamma}\cap \Delta_J^{(n,0)}$ into the strata $C_{\delta}\cap \sigma_{J,\gamma}\cap \Delta_J^{(n,0)}$. It is a fact that $\Phi_a=\Phi_b$ if and only if $a, b$ belong the same stratum. Now, to deduce the lemma, we use the same arguments as in the proof of~\cite[Proposition~3.13]{Raibaut}.
\end{proof}

In particular, when $\gamma_a^J=\Gamma$ we have $a=(0,\dots,0)$, $J=[d]$, and $f_{\gamma_a}(x)=f(x)$. Then the morphism $\Phi_a$ is nothing but the identity morphism.

Similarly, we also consider the morphism $\Psi_a: X_{J,\gamma}(0)\to X_0$ sending $(x_i)_{i\in J}$ to $(\hat x_1,\dots,\hat x_d)$, which commutes with $\pi$ in Theorem~\ref{mainthm1} and the morphism $\varphi(t)\mapsto \varphi(0)$ for $a\in \mathring{\sigma}_{J,\gamma}\cap \Delta_J^{(n,k)}$ and $k\geq 1$. Recall that the $\mu_n$-action on $X_{J,\gamma}(0)$ is trivial. As above, we also have

\begin{lemm}
If $a, b\in \mathring{\sigma}_{J,\gamma}\cap \Delta_J^{(n,0)}$, then $[\Psi_a: X_{J,\gamma}(0)\to X_0]=[\Psi_b: X_{J,\gamma}(0)\to X_0]$ in $\mathscr M_{X_0}^{\mu_n}$.
\end{lemm}

Notice that, for $J\subseteq [d]$ and $I\subseteq J$, we identify $(\mathbb R_{>0})^I$ with the set of $(x_j)_{j\in J}$ in $(\mathbb R_{\geq 0})^J$ such that $x_i>0$ for all $i\in I$ and $x_j=0$ for all $j\in J\setminus I$.

For $0\leq m\leq d$, denote by $F(m)$ the set of $\gamma\in F$ such that for all $J\supseteq J_{\gamma}$ and $\gamma=\gamma_a^J$ we have $a_i\geq 1$ for all $i\in J\cap [m+1,d]$. Note that $F(0)=K$ and $F(d)=F$.

The last part of the following result is known in~\cite{G,AB-CN-L-MH,GLM2}, we provide new formulas in i), ii) as follows.

\begin{theo}\label{thm4.1}
Let $f$ be in $\mathbb C[x_1,\dots,x_d]$ with $f(O)=0$, let $d_1, d_2$ be in $\mathbb N$ with $d=d_1+d_2$. For any $\gamma\in F$, put $\Lambda_{\gamma}:=\{I\subseteq J_{\gamma} \,|\, \mathring{\sigma}_{J_{\gamma},\gamma}\cap (\mathbb R_{>0})^I\not=\emptyset\}$ and
\[
\lambda_{\gamma}:=\sum_{I\in \Lambda_{\gamma}}(-1)^{\dim(\mathring{\sigma}_{J_{\gamma},\gamma}\cap (\mathbb R_{>0})^I)}.
\]
(Hence, as $\gamma\in K$, $\lambda_{\gamma}=(-1)^{\dim (\mathring{\sigma}_{J_{\gamma},\gamma})}=(-1)^{|J_{\gamma}|-\dim(\gamma)}$.) The below identities hold in $\mathscr{M}_{X_0}^{\hat\mu}$ for \eqref{12i}, in $\mathscr{M}_{\mathbb A_{\mathbb C}^{d_1}}^{\hat\mu}$ for \eqref{12ii}, and in $\mathscr{M}_{\mathbb C}^{\hat\mu}$ for \eqref{12iii}.
\begin{enumerate}\romanenumi
\item \label{12i} If $f$ is Newton nondegenerate, then
\begin{align*}
\mathscr{S}_f=-\sum_{\gamma\in F\setminus \widetilde{F}}\lambda_{\gamma}\left[X_{\gamma}(1)\to {X_0}\right]+\sum_{\gamma\in F}\lambda_{\gamma}\left[X_{\gamma}(0)\to {X_0}\right].
\end{align*}
\item \label{12ii} If $f$ is Newton nondegenerate and $\iota: \mathbb A_{\mathbb C}^{d_1}\equiv \mathbb A_{\mathbb C}^{d_1}\times_{\mathbb C}\{0\}^{d_2}\hookrightarrow X_0$ is an inclusion, then
\begin{align*}
\iota^*\mathscr{S}_f=-\sum_{\gamma\in F(d_1)\setminus \widetilde{F}}\lambda_{\gamma}\left[X_{\gamma}(1)\times_{X_0}\mathbb A_{\mathbb C}^{d_1}\to \mathbb A_{\mathbb C}^{d_1}\right]+\sum_{\gamma\in F(d_1)}\lambda_{\gamma}\left[X_{\gamma}(0)\times_{X_0}\mathbb A_{\mathbb C}^{d_1}\to \mathbb A_{\mathbb C}^{d_1}\right].
\end{align*}
\item \label{12iii} If $f$ is nondegenerate in the sense of Kouchnirenko, then
\[
\mathscr{S}_{f,O}=\sum_{\gamma\in K}(-1)^{|J_{\gamma}|+1-\dim(\gamma)}\left([X_{\gamma}(1)]-[X_{\gamma}(0)]\right).
\]
\end{enumerate}
\end{theo}

\begin{proof}
Notice that \eqref{12i} is not a particular case of \eqref{12ii} in general, but our proof method of \eqref{12i} is similar to that of \eqref{12ii}; while \eqref{12iii} is really a consequence of \eqref{12ii} (when $d_1=0$); so it suffices to prove \eqref{12ii}. By the decomposition~\eqref{decompXn} and Lemma~\ref{indexPn}$\MK$\eqref{8i}, we have
\begin{align*}
\mathscr X_{n}(f)=\bigsqcup_{\gamma\in F}\ \bigsqcup_{J\supseteq J_{\gamma}} \ \bigsqcup_{k\in \mathbb N}\ \bigsqcup_{a\in \mathring{\sigma}_{J,\gamma}\cap \Delta_J^{(n,k)}}\mathscr{X}_{J,a}^{(n)}.
\end{align*}
Take the fiber product on both sides with $\iota: \mathbb A_{\mathbb C}^{d_1}\hookrightarrow X_0$. If there is an $i\in J\cap [d_1+1,\dots,d]$ with $a_i=0$ (i.e. $\gamma\not\in F(d_1)$), then $\varphi_i(0)$ is in $\mathbb G_m$, thus the image of $\iota$ is disjoint with the image of $\mathscr{X}_{J,a}^{(n)}$ in $X_0$, so $\iota^*[\mathscr{X}_{J,a}^{(n)}]=0$. It follows that
\[
\iota^*[\mathscr X_{n}(f)]=\sum_{\gamma\in F(d_1)}\ \sum_{J\supseteq J_{\gamma}} \ \sum_{k\in \mathbb N}\ \sum_{a\in \mathring{\sigma}_{J,\gamma}\cap \Delta_J^{(n,k)}}\iota^*\bigl[\mathscr{X}_{J,a}^{(n)}\bigr].
\]
Using Lemma~\ref{coordinateface}, the diagram~\eqref{eq3.5} for $X_{J,\gamma}(1)\to X_0$ and a similar one for $X_{J,\gamma}(0)\to X_0$ we have
\begin{align*}
&\sum_{n\geq 1}\iota^*\bigl[\mathscr X_n(f)\bigr]\LL^{-nd}T^n\\
&\qquad=\sum_{\gamma\in F(d_1)\setminus \widetilde{F}}\sum_{J\supseteq J_{\gamma}}\bigl[X_{J,\gamma}(1)\times_{X_0}\mathbb A_{\mathbb C}^{d_1}\to \mathbb A_{\mathbb C}^{d_1}\bigr]S_{J,\gamma}^0(T)+ \sum_{\gamma\in F(d_1)}\sum_{J\supseteq J_{\gamma}} \bigl[X_{J,\gamma}(0)\times_{X_0}\mathbb A_{\mathbb C}^{d_1}\to \mathbb A_{\mathbb C}^{d_1}\bigr]S_{J,\gamma}^{>}(T),
\end{align*}
where
\[
S_{J,\gamma}^0(T)=\sum_{\substack{ a\in \mathring{\sigma}_{J,\gamma}\cap \mathbb N^J\\ a_j\leq \ell_J(a), j\in J}}\LL^{(|J|-d)\ell_J(a)-|a|}T^{\ell_J(a)}
\]
and
\[
S_{J,\gamma}^{>}(T)=\sum_{k\geq 1}\sum_{\substack{a\in \mathring{\sigma}_{J,\gamma}\cap \mathbb N^J\\ a_j\leq \ell_J(a)+k, j\in J}}\LL^{(|J|-d)(\ell_J(a)+k)-|a|-k}T^{\ell_J(a)+k}.
\]
The conclusion then follows from Lemma~\ref{lem37}.
\end{proof}

We need the following lemmas.

\begin{lemm}\label{intersectionlem}
If $\gamma\in F$, $J\supsetneqq J_{\gamma}$ and $I\subseteq J_{\gamma}$, then $\mathring{\sigma}_{J,\gamma}\cap (\mathbb N^*)^I=\emptyset$.
\end{lemm}

\begin{proof}
We first claim that if $a\in \sigma_{J, \gamma}$ and $j\in J\setminus J_{\gamma}$ then for all $t\geq 0$ we have $a+te^j\in \sigma_{J, \gamma}$, where $e^j$ is defined in the proof of Lemma~\ref{coordinateface}. Indeed, for any $b\in \gamma$, we have
\[
\langle a+te^j, b \rangle = \langle a, b \rangle\leq \langle a, c \rangle\leq \langle a+te^j, c \rangle\quad \text{for all}\ \ c\in \Gamma(f^J).
\]
That implies $\langle a+te^j, b \rangle= \ell_J(a+te^j)$. Hence $a+te^j\in \sigma_{J, \gamma}$.

We assume by contradiction that there exists some point $a\in \mathring{\sigma}_{J,\gamma}\cap (\mathbb N^*)^I.$ Take $m\in J\setminus J_{\gamma}$. One can write the cone ${\sigma}_{J,\gamma}$ as
\[
\sigma_{J, \gamma}= \left\{\alpha\in \mathbb R^J \,\middle|\, h_i(\alpha)=0, k_j(\alpha)\geq 0, l_s(\alpha)>0, i\in I_1, j\in I_2, s\in I_3\right\},
\]
where $h_i$, $k_j$, $l_s$ are linear forms on $\mathbb{R}^J$. Since $a\in \mathring{\sigma}_{J,\gamma}$ we get $k_j(a)>0$ for all $j\in I_2$. Hence, for $t>0$ small, we get $k_j(a-te^m)>0$. Similarly, we get $l_s(a-te^m)>0$ for $t>0$ small and for all $s\in I_3$. By the above argument, for $t>0$ we have $a+te^m\in \sigma_{J, \gamma}$, hence $h_i(a+te^m)= h_i(a)=0$, so $h_i(a-te^m)=0$, for every $i\in I_1$. It implies that $a-te^m\in \sigma_{J, \gamma}$ for $t>0$ small. On the other hand, because $m\in J\setminus J_{\gamma}$, $I\subseteq J_{\gamma}$ and $a\in (\mathbb N^*)^I$, we have $a_m=0$, so $a-te^m\notin \mathbb{R}_{\geq 0}^J$ for any $t>0$. This is a contradiction, and the lemma is proved.
\end{proof}

\begin{lemm}\label{lem37}
Use the notation in Theorem~\ref{thm4.1} and its proof, and let $\gamma\in F$. If $J\supsetneqq J_{\gamma}$, then
\[
\lim_{T\to\infty}S_{J,\gamma}^0(T)=\lim_{T\to\infty}S_{J,\gamma}^{>}(T)=0.
\]
If $J=J_{\gamma}$, then
\[
\lim_{T\to\infty}S_{J_{\gamma},\gamma}^0(T)=-\lim_{T\to\infty}S_{J_{\gamma},\gamma}^{>}(T)=\lambda_{\gamma}.
\]
\end{lemm}

\begin{proof}
Assume that $J\supsetneqq J_{\gamma}$. Because
\begin{itemize}
\item $(\mathbb R_{\geq 0})^J=\bigsqcup_{I\subseteq J}(\mathbb R_{>0})^I$, where $(\mathbb R_{>0})^{\emptyset}=\{(0,\dots,0)\}$ by convention,
\item $\ell_J(a)=\ell_{J_{\gamma}}(a)$ for $a\in \mathring{\sigma}_{J,\gamma}\cap \mathbb N^J$,
\item if $I\subseteq J_{\gamma}$, then $\mathring{\sigma}_{J,\gamma}\cap (\mathbb N^*)^I=\emptyset$ (by Lemma~\ref{intersectionlem}),
\end{itemize}
we have $S_{J,\gamma}^0(T)=\sum_{I\subseteq J, I\not\subseteq J_{\gamma}}S_{J,I,\gamma}^0(T)$, where
\[
S_{J,I,\gamma}^0(T):=\sum_{
\begin{smallmatrix} a\in \mathring{\sigma}_{J,\gamma}\cap (\mathbb N^*)^I\\ a_j\leq \ell_{J_{\gamma}}(a), j\in J
\end{smallmatrix}}\LL^{-|a|}(\LL^{|J|-d}T)^{\ell_{J_{\gamma}}(a)}=\sum_{
\begin{smallmatrix} a\in \mathring{\sigma}_{J,\gamma}\cap (\mathbb N^*)^I\\ a_j\leq \ell_{J_{\gamma}}(a), j\in I\setminus J_{\gamma}
\end{smallmatrix}}\LL^{-|a|}(\LL^{|J|-d}T)^{\ell_{J_{\gamma}}(a)}
\]
(the last equality comes because the inequality $a_j\leq \ell_{J_{\gamma}}(a)$ is automatic for every $j\in J_{\gamma}$). Denote by $H_j$ the half space of $\mathbb R^I$ defined by $a_j\leq \ell_{J_{\gamma}}(a)$. Then $\mathring{\sigma}_{J,\gamma}\cap (\mathbb R_{>0})^I\not\subseteq H_j$ for all $j\in I\setminus J_{\gamma}$, because if there exists an $a\in \sigma_{J, \gamma}\cap H_j$, then for $t>0$ large enough, we have $a+te^j\in \sigma_{J, \gamma}$ but $a+te^j\not\in H_j$. This agrees with the hypothesis of Lemma~\ref{lem21}. Hence, by Lemma~\ref{lem21}, $\lim_{T\to\infty}S_{J,I,\gamma}^0(T)=0$ for any $I\subseteq J$ and $I\not\subseteq J_{\gamma}$. Hence $\lim_{T\to \infty}S_{J,\gamma}^0(T)=0$. Similarly, we have $\lim_{T\to \infty}S_{J,\gamma}^{>}(T)=0$.

For the rest statement, it follows from~\cite[Lemme~2.1.5]{G} that
\begin{align*}
\lim_{T\to \infty}S_{J_{\gamma},\gamma}^0(T)=\sum_{I\subseteq J_{\gamma}}\lim_{T\to \infty}\sum_{a\in \mathring{\sigma}_{J_{\gamma},\gamma}\cap (\mathbb N^*)^I}\LL^{-|a|}(\LL^{|J|-d}T)^{\ell_{J_{\gamma}}(a)}=\lambda_{\gamma},
\end{align*}
and similarly, $\lim_{T\to \infty}S_{J_{\gamma},\gamma}^{>}(T)=-\lambda_{\gamma}$.
\end{proof}


\begin{rema}
This result revisits Guibert's work in~\cite[Section~2.1]{G} for Newton nondegenerate polynomials $f$ in a more general setting. Indeed, in~\cite{G} Guibert requires $f$ to have the form $\sum_{\alpha\in (\mathbb N^*)^d}a_{\alpha}x^{\alpha}$, while we do not. Recently, Bultot--Nicaise in~\cite[Theorems~7.3.2, 7.3.5]{BN} provide a new approach to the motivic zeta functions $Z_f(T)$ and $Z_{f,O}(T)$, for $f$ being Newton nondegenerate, using log smooth models.
\end{rema}

\begin{exem}
Consider the function $f(x,y)=y^2-x^3$ on $\mathbb A_{\mathbb C}^2$, which is well known to be nondegenerate with respect to its Newton polyhedron $\Gamma$. If $\gamma$ is either the face $[3,+\infty)\times\{0\}$ or the face $\{0\}\times [2,+\infty)$ of $\Gamma$, then $\gamma$ is a coordinate face and $X_{\gamma}(0)=\emptyset$. If $\gamma$ is the compact face $\{(3,0)\}$ (resp. $\{(0,2)\}$), it contributes $X_{\gamma}(1)=\mu_3$ and $X_{\gamma}(0)=\emptyset$ (resp. $X_{\gamma}(1)=\mu_2$ and $X_{\gamma}(0)=\emptyset$), as well as $\lambda_{\gamma}=-1$. If $\gamma$ is the compact face connecting $(3,0)$ and $(0,2)$, it contributes $X_{\gamma}(1)=\bigl\{(x,y)\in \mathbb G_{m,\mathbb C}^2\,\bigm|\, y^2-x^3=1\bigr\}$ and $X_{\gamma}(0)=\bigl\{(x,y)\in \mathbb G_{m,\mathbb C}^2\,\bigm|\, y^2-x^3=0\bigr\}\cong \mathbb G_{m,\mathbb C}$, as well as $\lambda_{\gamma}=-1$. Finally, if $\gamma=\Gamma$, it is a coordinate face and contributes $X_{\Gamma}(0)\cong \mathbb G_{m,\mathbb C}$, as well as $\lambda_{\Gamma}=1$ (since $\mathring{\sigma}_{\{1,2\},\Gamma}=\{(0,0)\}$). By Theorem~\ref{thm4.1} (we skip arrows to $X_0$ for simplicity),
\begin{align*}
\mathscr S_f&=\bigl[\bigl\{(x,y)\in \mathbb G_{m,\mathbb C}^2\,\bigm|\, y^2-x^3=1\bigr\}\bigr]+[\mu_3]+[\mu_2]-(\LL-1)+(\LL-1)\\
&=\bigl[\bigl\{(x,y)\in \mathbb A_{\mathbb C}^2\,\bigm|\, y^2-x^3=1\bigr\}\bigr] \quad (\in \mathscr M_{X_0}^{\hat\mu}).
\end{align*}
This also agrees with Davison--Meinhardt's conjecture on motivic nearby fibers of weighted homogeneous polynomials mentioned in~\cite{NP}. Also by Theorem~\ref{thm4.1} we have
\begin{align*}
\mathscr S_{f,O}&=\bigl[\bigl\{(x,y)\in \mathbb G_{m,\mathbb C}^2\,\bigm|\, y^2-x^3=1\bigr\}\bigr]+[\mu_3]+[\mu_2]-(\LL-1)\\
&=\bigl[\bigl\{(x,y)\in \mathbb A_{\mathbb C}^2\,\bigm|\, y^2-x^3=1\bigr\}\bigr]-(\LL-1) \quad (\in \mathscr M_{\mathbb C}^{\hat\mu}).
\end{align*}
\end{exem}

\subsection{Relation between motivic nearby cycles of \texorpdfstring{$f$}{f} and \texorpdfstring{$f^{[d-1]}$}{f[d-1]}}

Let $w$ be a linear function on $\mathbb C^d$ generic to $f$. In~\cite{LDT75, LDT77}, L\^e D\~ung Tr\'ang introduced the relative monodromy concerning $(f,O)$ and $w$. We refer to~\cite[Theorem~2.4]{LDT77} for the following. Denote by $B_{\varepsilon}$ the closed $d$-balls of radius $\varepsilon$ about $O$, by $D_{\eta}$ the closed disk of radius $\eta$ about $0$ in $\mathbb C$, and by $D_{\eta}^{\times}$ the punctured disk $D_{\eta}\setminus\{0\}$. Let $\Phi$ be the restriction of the map $(w,f): \mathbb C^d\to \mathbb C^2$ to $B_{\epsilon}\cap (w,f)^{-1}(D_{\eta}^2)$. L\^e proved that, for $0<\eta\ll\epsilon\ll 1$, the map $\Phi^{-1}(D_{\eta}^2\setminus (D_{\eta}\times \{0\}))\to D_{\eta}^{\times}$ is a smooth fibration which is fiber isomorphic to the Milnor fibration of $(f,O)$ with monodromy $M: \Phi^{-1}(D_{\eta}\times \{\eta\})\to \Phi^{-1}(D_{\eta}\times \{\eta\})$, and that $w^{-1}(0)\cap\Phi^{-1}(D_{\eta}^2\setminus (D_{\eta}\times \{0\}))\to D_{\eta}^{\times}$ is also a smooth fibration which is fiber isomorphic to the Milnor fibration of $(f|_{w=0},O)$. Then $M$ induces the monodromy of the Milnor fibration of $(f|_{w=0},O)$ and it lifts a diffeomorphism (which is a carousel) $D_{\eta}\times\{\eta\}\to D_{\eta}\times\{\eta\}$ along the mapping $\Phi|_{\Phi^{-1}(D_{\eta}\times \{\eta\})}$.

We now consider the ``non-generic'' hyperplane $x_d=0$. We write $\widetilde f$ for $f^{[d-1]}$, that is,
\[
\widetilde f(x_1,\ldots,x_{d-1})=f(x_1,\ldots,x_{d-1},0),
\]
and write $\widetilde O$ for the origin of $\mathbb C^{d-1}$. Let $\widetilde X_0$ be the zero locus of $\widetilde f$, which may be included into $X_0$. The following theorem may be partially considered as a motivic analogue of the work mentioned above. Using realizations, it would be interesting to compare the motivic result to the topological result of L\^e D\~ung Tr\'ang.

\goodbreak
\begin{theo}\label{restricting}
Let $f\in \mathbb C[x_1,\dots,x_d]$, and let $d_1, d_2 \in\mathbb N$ such that $d=d_1+d_2$. The below identities hold in $\mathscr{M}_{\mathbb A_{\mathbb C}^{d_1}}^{\hat\mu}$ for \eqref{17i} and in $\mathscr{M}_{\mathbb C}^{\hat\mu}$ for~\eqref{17ii}.
\begin{enumerate}\romanenumi
\item \label{17i} Suppose that $f$ is Newton nondegenerate, $\widetilde X_0\subseteq X_0$ and that $\mathbb A_{\mathbb C}^{d_1}$ is embedded into $\widetilde X_0$ with the inclusions of $\mathbb A_{\mathbb C}^{d_1}$ in both $X_0$ and $\widetilde X_0$ denoted by the same symbol $\iota$. Then
\[
\iota^*\mathscr{S}_f=\iota^*\mathscr{S}_{\widetilde{f}}+\iota^*\mathscr S_{f,x_d}^{\Delta}.
\]
\item \label{17ii} If $f$ is nondegenerate in the sense of Kouchnirenko and $f(O)=0$, then
\[
\mathscr{S}_{f,O}=\mathscr{S}_{\widetilde f,\widetilde O}+\mathscr S_{f,x_d,O}^{\Delta}.
\]
\end{enumerate}
\end{theo}

\begin{proof}
It suffices to prove \eqref{17i}. By the definition of $(n,m)$-iterated contact loci, we have
\begin{align*}
\mathscr{X}_{n,m}(f,x_d)=\bigsqcup_{(J,a)\in \mathcal P_n. a_d=m}\mathscr X_{J_{\gamma},a}^{(n)},
\end{align*}
We deduce from Section~\ref{section2.3} and the method in the proof of Theorem~\ref{thm4.1} that
\begin{multline*}
\sum_{n\geq m\geq 1}\iota^*[\mathscr{X}_{n,m}(f,x_d)]\LL^{-nd}T^n\\
=\sum_{\gamma\in F(d_1)}\sum_{J\supseteq J_{\gamma}}\sum_{\substack{a\in \mathring{\sigma}_{J,\gamma}\cap \mathbb N^J\\ a_j\leq \ell_J(a), j\in J\\ 1\leq a_d\leq \ell_J(a)}}\iota^*\bigl[\mathscr X_{J,a}^{(\ell_J(a))}\bigr]\LL^{-d\ell_J(a)}T^{\ell_J(a)}\\
+\sum_{\gamma\in F(d_1)}\sum_{J\supseteq J_{\gamma}}\sum_{k\geq 1}\sum_{\substack{a\in \mathring{\sigma}_{J,\gamma}\cap \mathbb N^J\\ a_j\leq \ell_J(a), j\in J\\ 1\leq a_d\leq \ell_J(a)+k}}\iota^*\bigl[\mathscr X_{J,a}^{(\ell_J(a)+k)}\bigr]\LL^{-d(\ell_J(a)+k)}T^{\ell_J(a)+k}.
\end{multline*}
We apply Theorem~\ref{mainthm1} to $\gamma\in F(d_1)$ and $a\in \mathring{\sigma}_{J,\gamma}$. If $d\in J_{\gamma}$, then $\ell_{J_{\gamma}}(a)+k\geq a_d\geq 1$ automatically for any $k\in \mathbb N$. If $d\not\in J_{\gamma}$, then the inequalities $\ell_{J_{\gamma}}(a)+k\geq a_d\geq 1$ is in the situation of~\cite[Lemma~2.10]{GLM1}, in which the corresponding series has the limit zero. Therefore, taking $\lim_{T\to \infty}$ and using Theorem~\ref{mainthm1}, Lemma~\ref{lem37} and the proof of Theorem~\ref{thm4.1}, we get
\begin{align*}
\iota^*\mathscr S_{f,x_d}^{\Delta}=-\sum_{\substack{\gamma\in F(d_1)\setminus \widetilde{F}\\ d\in J_{\gamma}}}\lambda_{\gamma}\bigl[X_{\gamma}(1)\times_{X_0}\mathbb A_{\mathbb C}^{d_1}\to \mathbb A_{\mathbb C}^{d_1}\bigr]+\sum_{\substack{\gamma\in F(d_1)\\ d\in J_{\gamma}}}\lambda_{\gamma}\bigl[X_{\gamma}(0)\times_{X_0}\mathbb A_{\mathbb C}^{d_1}\to \mathbb A_{\mathbb C}^{d_1}\bigr].
\end{align*}
By Theorem~\ref{thm4.1},
\[
\iota^*\mathscr S_{f}=-\!\!\sum_{\substack{\gamma\in F(d_1)\setminus \widetilde{F}\\ d\not\in J_{\gamma}}}\!\!\lambda_{\gamma}\bigl[X_{\gamma}(1)\times_{X_0}\mathbb A_{\mathbb C}^{d_1}\to \mathbb A_{\mathbb C}^{d_1}\bigr]+\!\sum_{\substack{\gamma\in F(d_1)\\ d\not\in J_{\gamma}}}\!\!\lambda_{\gamma}\bigl[X_{\gamma}(0)\times_{X_0}\mathbb A_{\mathbb C}^{d_1}\to \mathbb A_{\mathbb C}^{d_1}\bigr]+\iota^*\mathscr S_{f,x_d}^{\Delta}.
\]
The condition $d\not\in J_{\gamma}$ means that $J_{\gamma}\subseteq [d-1]$, hence, again by Theorem~\ref{thm4.1},
\[
\iota^*\mathscr S_{\widetilde f}=-\!\!\sum_{\substack{\gamma\in F(d_1)\setminus \widetilde{F}\\ d\not\in J_{\gamma}}}\!\!\lambda_{\gamma}\bigl[X_{\gamma}(1)\times_{X_0}\mathbb A_{\mathbb C}^{d_1}\to \mathbb A_{\mathbb C}^{d_1}\bigr]+\!\sum_{\substack{\gamma\in F(d_1)\\ d\not\in J_{\gamma}}}\!\!\lambda_{\gamma}\bigl[X_{\gamma}(0)\times_{X_0}\mathbb A_{\mathbb C}^{d_1}\to \mathbb A_{\mathbb C}^{d_1}\bigr].
\]
The theorem is completely proved.
\end{proof}


\section{Cohomology groups of contact loci of nondegenerate singularities}\label{Sec5-cloci}

As before, let $f$ be in $\mathbb C[x_1,\dots,x_d]$ which vanishes at $O$. In this section, we always assume that $f$ is nondegenerate in the sense of Kouchnirenko (say for short that $f$ is \emph{nondegenerate}).

\subsection{Borel--Moore homology groups of contact loci}

Consider the decomposition of $\mathscr X_{n,O}(f)$ shown in~\eqref{otherdecomp} and Lemma~\ref{indexPn}$\MK$\eqref{8ii}:
\[
\mathscr X_{n,O}(f)=\bigsqcup_{(J,a)\in \widetilde{\mathcal P}_n}\mathscr{X}_{J,a}^{(n)},
\]
where $\widetilde{\mathcal P}_n$ is the set of all the pairs $(J,a)$ such that $J\supseteq J_{\gamma}$, $a\in \bigsqcup_{k\in \mathbb N}\big(\mathring{\sigma}_{J,\gamma}\cap \widetilde\Delta_J^{(n,k)}\big)$ and $\gamma\in K$. We consider an ordering in $\widetilde{\mathcal P}_n$ defined as follows: for $(J, a)$ and $(J', a')$ in $\widetilde{\mathcal P}_n$, $(J', a')\leq (J, a)$ if and only if $J'\subseteq J$ and $a_j\leq a'_j$ for all $j\in J'$, where $a=(a_j)_{j\in J}$ and $a'=(a'_j)_{j\in J'}$.

\begin{lemm}\label{lem5.1}
Let $n$ be in $\mathbb N^*$. For all $(J, a)$ and $(J', a')$ in $\widetilde{\mathcal P}_n$ such that $\mathscr{X}_{J',a'}^{(n)}$ and $\mathscr X_{J,a}^{(n)}$ are nonempty, the following are equivalent:
\begin{enumerate}\romanenumi
\item \label{18i} $(J', a')\leq (J, a),$
\item \label{18ii} $\mathscr{X}_{J',a'}^{(n)}\subseteq \overline{\mathscr X_{J,a}^{(n)}},$
\item \label{18iii} $\mathscr{X}_{J',a'}^{(n)}\cap \overline{\mathscr X_{J,a}^{(n)}}\neq \emptyset,$
\end{enumerate}
the closure taken in the usual topology. Consequently, for all $(J,a)\in \widetilde{\mathcal P}_n$ such that $\mathscr X_{J,a}^{(n)}\neq \emptyset$ we have
\[
\overline{\mathscr X_{J,a}^{(n)}}= \bigsqcup_{(J', a')\leq (J, a)} \mathscr{X}_{J',a'}^{(n)}.
\]
\end{lemm}

\begin{proof}
We prove here that \eqref{18iii} implies \eqref{18i}, the rest are straighforward. Observe firstly that, due to the definition of $\mathscr X_{J,a}^{(n)}$, if it is nonempty then
\[
\overline{\mathscr{X}_{J,a}^{(n)}}=\left\{\varphi\in\mathscr{X}_n(f)\mid \ord_tx_j(\varphi)\geq a_j \ \forall j\in J,\ x_i(\varphi)\equiv 0\ \forall i\not\in J\right\},
\]
where $a=(a_j)_{j\in J}$, and $a_j>0$ for all $j\in J$. Assume that there exists $\varphi^0\in \mathscr{X}_{J',a'}^{(n)}\cap \overline{\mathscr X_{J,a}^{(n)}}\neq \emptyset$. Then we have $\ord_tx_j(\varphi^0)=a'_j>0$ for all $j\in J'$, and $\ord_tx_j(\varphi^0)\geq a_j>0$ for all $j\in J$. If $i\not\in J$, then we have $\ord_tx_i(\varphi^0)=+\infty$, thus $i\not\in J'$, so $J'\subseteq J$. Clearly, $a_j\leq a'_j$ for all $j\in J'$. Therefore, $(J', a')\leq (J, a)$.
\end{proof}

Consider the function $\eta: \widetilde{\mathcal P}_n\to \mathbb{Z}$ given by $\eta(J,a)=\dim_{\mathbb{C}}\mathscr{X}_{J,a}^{(n)}$, for every $n\in \mathbb N^*$. Put
\begin{align*}
S_p:= \bigsqcup_{(J, a)\in \widetilde{\mathcal P}_n, \eta(J,a)\leq p}\mathscr{X}_{J,a}^{(n)},
\end{align*}
for $p\in \mathbb N$. The below are some properties of $\eta$ and $S_p$'s.

\begin{lemm}\label{lem5.3}
Let $n$ be in $\mathbb N^*$.
\begin{enumerate}\romanenumi
\item \label{19i} If $(J', a')\leq (J, a)$ in $\widetilde{\mathcal P}_n$ and $\mathscr X_{J,a}^{(n)}\neq \emptyset$, then $\eta(J', a')\leq \eta(J, a)$.
\item \label{19ii} For all $p\in \mathbb N$, $S_p$ are closed and $S_p\subseteq S_{p+1}$. As a consequence, there is a filtration of $\mathscr X_{n,O}(f)$ by closed subspaces:
\[
\mathscr X_{n,O}(f)=S_{d_0}\supseteq S_{d_0-1}\supseteq \cdots \supseteq S_{-1}=\emptyset,
\]
where $d_0$ denotes the $\mathbb C$-dimension of $\mathscr X_{n,O}(f)$.
\end{enumerate}
\end{lemm}

\begin{proof}
The first statement \eqref{19i} is trivial. To prove \eqref{19ii} we take the closure of $S_p$; then using Lemma~\ref{lem5.1} we get
\[
\overline{S}_p=\bigcup_{\eta(J, a)\leq p} \overline{\mathscr X_{J,a}^{(n)}}=\bigcup_{\eta(J, a)\leq p, \mathscr X_{J,a}^{(n)}\neq \emptyset} \overline{\mathscr X_{J,a}^{(n)}}= \bigcup_{\eta(J, a)\leq p, \mathscr X_{J,a}^{(n)}\neq \emptyset} \bigsqcup_{(J',a')\leq (J, a)}\mathscr{X}_{J',a'}^{(n)}.
\]
This decomposition combined with \eqref{19i} implies that $\overline{S}_p\subseteq S_p$, which proves that $S_p$ is a closed subspace. The remaining statements of \eqref{19ii} are trivial.
\end{proof}

A main result of this section is the following theorem. To express the result, we work with the Borel--Moore homology $H_*^{\BM}$.

\begin{theo}\label{thm5.4}
Let $f\in \mathbb C[x_1,\dots,x_d]$ be nondegenerate in the sense of Kouchnirenko, $n\in \mathbb N^*$ and $f(O)=0$. Then there is a spectral sequence
\[
E_{p,q}^1:=\bigoplus_{(J, a)\in \widetilde{\mathcal P}_n,\eta(J, a)=p}H^{\BM}_{p+q}(\mathscr{X}_{J,a}^{(n)})\Longrightarrow H^{\BM}_{p+q}(\mathscr X_{n,O}(f)).
\]
\end{theo}

\begin{proof}
We have the following the Gysin exact sequence
\[
\cdots \to H^{\BM}_{p+q}(S_{p-1})\to H^{\BM}_{p+q}(S_{p})\to H^{\BM}_{p+q}(S_{p}\setminus S_{p-1})\to H^{\BM}_{p+q-1}(S_{p-1})\to \cdots.
\]
Put
\[
A_{p, q}:= H^{\BM}_{p+q}(S_{p}),\quad E_{p,q}:=H^{\BM}_{p+q}(S_{p}\setminus S_{p-1}).
\]
Then we have the bigraded $\mathbb{Z}$-modules $A:= \bigoplus_{p,q} A_{p,q}$ and $E:= \bigoplus_{p,q} E_{p,q}$. The previous exact sequence induces the exact couple $\langle A, E; h, i, j\rangle,$ where $h: A\to A$ is induced from the inclusions $S_m\subseteq S_{m+1}$, $i: A\to E$ and $j: E\to A$ are induced from the above exact sequence. Since the filtration in Lemma~\ref{lem5.3}$\MK$\eqref{19ii} is finite, that exact couple gives us the following spectral sequence
\[
E^1_{p,q}:=E_{p,q}=H^{\BM}_{p+q}(S_{p}\setminus S_{p-1})\Longrightarrow H^{\BM}_{p+q}(\mathscr X_{n,O}(f)).
\]

On the other hand, we have
\[
S_p\setminus S_{p-1}= \bigsqcup_{(J, a)\in \widetilde{\mathcal P}_n, \eta(J, a)=p} \mathscr{X}_{J,a}^{(n)}.
\]
One claims that for two different pairs $(J, a), (J^{'}, a^{'})$ in $\widetilde{\mathcal P}_n$ which $\eta(J, a)=\eta(J^{'}, a^{'})=p$ then
\[
\mathscr{X}_{J,a}^{(n)}\cap \overline{\mathscr X_{J^{'}, a^{'}}^{(n)}}= \emptyset\quad \text{and}\quad
\mathscr X_{J^{'}, a^{'}}^{(n)}\cap \overline{\mathscr{X}_{J,a}^{(n)}}= \emptyset.
\]
Indeed, if otherwise, suppose that
\[
\mathscr{X}_{J^{'}, a^{'}}^{(n)}\cap \overline{\mathscr X_{J,a}^{(n)}}\neq \emptyset.
\]
By Lemma~\ref{lem5.1}, we obtain that $\mathscr{X}_{J^{'}, a^{'}}^{(n)}\subseteq \overline{\mathscr X_{J,a}^{(n)}}$, but $\mathscr{X}_{J^{'}, a^{'}}^{(n)}$ and $\mathscr X_{J,a}^{(n)}$ are two disjoint smooth manifolds, then $\eta(J^{'}, a^{'}) < \eta(J, a)$. This is a contradiction.

Therefore, in the set $S_p\setminus S_{p-1}$ with the induced topology, each set $\mathscr{X}_{J,a}^{(n)}$ which $ \eta(J, a)=p$ is open, hence, is also closed. This implies that
\[
H^{\BM}_{p+q}(S_{p}\setminus S_{p-1})= \bigoplus_{(J, a)\in \widetilde{\mathcal P}_n,\eta(J, a)=p}H^{\BM}_{p+q}(\mathscr{X}_{J,a}^{(n)}).
\]
The theorem is then proved.
\end{proof}

\begin{coro}
With the hypothesis as in Theorem~\ref{thm5.4}, there is an isomorphism of groups
\[
H^{\BM}_{2d_0}(\mathscr X_{n,O}(f))\cong \mathbb{Z}^s,
\]
where $s$ is the number of connected components of $\mathscr X_{n,O}(f)$ which have the same complex dimension $d_0$ as $\mathscr X_{n,O}(f)$.
\end{coro}


\subsection{Sheaf cohomology groups of contact loci}

In this subsection, we are going to prove the following theorem.

\begin{theo}\label{thm5.6}
Let $f\in\mathbb C[x_1,\dots,x_d]$ be nondegenerate in the sense of Kouchnirenko, $n\in \mathbb N^*$ and $f(O)=0$. Let $\mathcal F$ be an arbitrary sheaf of abelian groups on $\mathscr X_{n,O}(f)$. Then, there is a spectral sequence
\begin{equation}\label{spseq5.6}
E^{p,q}_1:=\bigoplus_{(J, a)\in \widetilde{\mathcal P}_n,\eta(J, a)=p}H^{p+q}_c(\mathscr{X}_{J,a}^{(n)}, \mathcal{F})\Longrightarrow H^{p+q}_c(\mathscr X_{n,O}(f), \mathcal{F}).
\end{equation}
\end{theo}

\begin{proof}
We use the notation in Lemma~\ref{lem5.3}. For simplicity, we write $S$ for $S_{d_0}=\mathscr X_{n,O}(f)$. For any $0\leq p \leq d_0$, we put $S_p^{\circ}:= S_p\setminus S_{p-1},$ which is a $\mu_n$-invariant subset of $S$. Consider the inclusions $j_p: S^{\circ}_p \hookrightarrow S_p$, $k_p: S\setminus S_p \hookrightarrow S$ and $i_p: S_p\hookrightarrow S$. Put $\mathcal{F}_p:= (j_{p})_!(j_p)^{-1}(i_p)^{-1}\mathcal{F}$ and $F^p(\mathcal{F}):= (k_{p-1})_!(k_{p-1})^{-1}\mathcal{F}$ for every $p\geq 1$, with the convention $F^0(\mathcal{F}):= \mathcal{F}$. Then we have the exact sequences
\[
0\to F^{p+1}(\mathcal{F})\to F^p(\mathcal{F}) \quad\text{and}\quad 0\to (i_{p})_*\mathcal{F}_p\to \mathcal{F}|_{S_p}\to \mathcal{F}|_{S_{p-1}},
\]
in which by $\mathcal{F}|_{S_p}$ we mean $(i_p)_*(i_p)^{-1}\mathcal F$. Therefore we have the following diagram
%\[
%\left.
%\begin{array}{ccccccccc}
%& & & & & & 0 & & \\
%& & & & & & \downarrow & & \\
%& & 0 & & & & (i_{p})_*\mathcal{F}_p & & \\
%& & \downarrow & & & & \downarrow & & \\
%0 & \to & F^{p+1}(\mathcal{F}) & \to & \mathcal{F} & \to & \mathcal{F}|_{S_p} & \to & 0 \\
%& & \downarrow & & || & & \downarrow & & \\
%0 & \to & F^{p}(\mathcal{F}) & \to & \mathcal{F} & \to & \mathcal{F}|_{S_{p-1}} & \to & 0
%\end{array}
%\right.
%\]
\[
\xymatrix@C=10pt@R=10pt{&&& 0\ar[d]\\
& 0 \ar[d] & & (i_{p})_*\mathcal{F}_p \ar[d]\\
0 \ar[r] & F^{p+1}(\mathcal{F}) \ar[r] \ar[d] & \mathcal{F} \ar[r] \ar@{=}[d] & \mathcal{F}|_{S_p} \ar[r] \ar[d] & 0\\
0 \ar[r] & F^{p}(\mathcal{F}) \ar[r] & \mathcal{F} \ar[r] & \mathcal{F}|_{S_{p-1}} \ar[r] & 0
}
\]
It implies from the snake lemma that $F^{p}(\mathcal{F})/F^{p+1}(\mathcal{F})\cong (i_{p})_*\mathcal{F}_p.$ Thus there is a filtration of $\mathcal{F}$ by ``skeleta'': $\mathcal{F}= F^{0}(\mathcal{F})\supseteq F^{1}(\mathcal{F})\supseteq \cdots.$ It gives the following spectral sequence of cohomology groups with compact support
\begin{equation}\label{eq:specseqYdt}
E_1^{p,q}(S,\mathcal{F}):= H^{p+q}_c(S,(i_{p})_*\mathcal{F}_p)
\Longrightarrow H^{p+q}_c(S,\mathcal{F}).
\end{equation}

Since $S_p$ is a closed subset of $S$, $H^{m}_c(S,(i_{p})_*\mathcal{F}_p)\cong H^{m}_c(S_p,\mathcal{F}_p)$ for any $m$ in $\mathbb N$. Also, by the isomorphisms given by the extension by zero sheaf, we have
\[
H^{m}_c(S_p,\mathcal{F}_p)= H^{m}_c(S_p,(j_{p})_!(j_p)^{-1}(i_p)^{-1}\mathcal{F}) \cong H^m_c(S_p^{\circ}, (j_p)^{-1}(i_p)^{-1}\mathcal{F}).
\]
We have that
\[
S_p^{\circ}=\bigsqcup_{(J, a)\in \widetilde{\mathcal P}_n,\eta(J,a)=p} \mathscr{X}_{J,a}^{(n)}.
\]
Then by the reason as in the proof of Theorem~\ref{thm5.4}, we get
\[
H^m_c(S_p^{\circ}, (j_p)^{-1}(i_p)^{-1}\mathcal{F})=\bigoplus_{(J, a)\in \widetilde{\mathcal P}_n,\eta(J, a)=p}H^{p+q}_c(\mathscr{X}_{J,a}^{(n)},(l_{J,a})^{-1}(j_p)^{-1}(i_p)^{-1}\mathcal{F}),
\]
where $l_{J,a}$ is the inclusion of $\mathscr{X}_{J,a}^{(n)}$ in $S_p^{\circ}$. For simplicity of notation, we write $H^{p+q}_c(\mathscr{X}_{J,a}^{(n)}, \mathcal{F})$ instead of $H^{p+q}_c(\mathscr{X}_{J,a}^{(n)},(l_{J,a})^{-1}(j_p)^{-1}(i_p)^{-1}\mathcal{F})$. The proof is completed.
\end{proof}

Now, let us consider the spectral sequence~\eqref{spseq5.6} for a constant sheaf. We need some notation, for each $\gamma\in K$, $k\in \mathbb N, n\in \mathbb N^*$, $p\in \mathbb Z$ and $J\supseteq J_{\gamma}$, we denote by $D_{J,\gamma,k,p}^{(n)}$ the set of all $a\in \mathring{\sigma}_{J,\gamma}\cap \widetilde\Delta^{(n,k)}_J$ such that $\dim_{\mathbb{C}}\mathscr{X}_{J,a}^{(n)}=p$.

\begin{lemm}
For any $\gamma\in K$, $k\in \mathbb N, n\in \mathbb N^*$, $p\in \mathbb Z$ and $J\supseteq J_{\gamma}$, the set $D_{J,\gamma,k,p}^{(n)}$ is finite.
\end{lemm}

\begin{proof}
Notice that $\dim_{\mathbb{C}}\mathscr{X}_{J,a}^{(n)}=d-1+|J|n-|a|-k$. The finiteness of $D_{J,\gamma,k,p}^{(n)}$ follows from the fact that the system of equations $d-1+|J|n-|a|-k=p$, $\ell_J(a)+k=n$ in variables $a$ only has finite solutions in $\mathbb N^J$.
\end{proof}

The summands in the spectral sequence~\eqref{spseq5.6} are described more explicitly in case of constant sheaf as below.

\begin{lemm}\label{lem46}
Let $\gamma\in K, n\in \mathbb N^*$, $p, q\in \mathbb Z$ and $J\supseteq J_{\gamma}$. Then, for any $a\in D_{J,\gamma,0,p}^{(n)}$ we have
\begin{equation*}
H^{p+q}_c(\mathscr{X}_{J,a}^{(n)}, \mathbb{C}) \cong H_{p-q}(X_{J,\gamma}(1), \mathbb{C}).
\end{equation*}
\end{lemm}

\begin{proof}
Since $a\in D_{J,\gamma,0,p}^{(n)}$, it follows from Theorem~\ref{mainthm1} that $\mathscr{X}_{J,a}^{(n)}$ is a complex manifold of real dimension~$2p$ and is homeomorphic to $X_{J,\gamma}(1) \times{\mathbb C}^{|J|\ell_J(a)-|a|}$. Then, by combining the duality and the Kunneth formula we get the conclusion.
\end{proof}

We also have the following description for the cohomology of $\mathscr{X}_{J,a}^{(n)}$ for $J\supseteq J_{\gamma}$ and $a\in D_{J,\gamma,k,p}^{(n)}$ with $k\in \mathbb N^*.$

\begin{lemm}\label{lem47}
Let $\gamma\in K, n, k\in \mathbb N^*$, $p, q\in \mathbb Z$ and $J\supseteq J_{\gamma}$. Then, for any $a\in D_{J,\gamma,k,p}^{(n)}$ we have
\begin{equation*}
H^{p+q}_c(\mathscr{X}_{J,a}^{(n)}, \mathbb{C}) \cong H_{p-q}(X_{J,\gamma}(0), \mathbb{C}).
\end{equation*}
\end{lemm}

\begin{proof}
Since $\mathscr{X}_{J,a}^{(n)}$ is a complex manifold of real dimension~$2p$, then by duality, we have
\[
H^{p+q}_c(\mathscr{X}_{J,a}^{(n)}, \mathbb{C}) \cong H_{p-q}(\mathscr{X}_{J,a}^{(n)}, \mathbb{C}).
\]
On the other hand, by Theorem~\ref{mainthm1}, $\mathscr{X}_{J,a}^{(n)}$ is a locally trivial fibration on $X_{J,\gamma}(0)$ with fiber ${\mathbb C}^{|J|(\ell_J(a)+k)-|a|-k}$ which is contractible. Hence, by the spectral sequence for (Serre) fibration, we obtain that $H_{p-q}(\mathscr{X}_{J,a}^{(n)}, \mathbb{C})\cong H_{p-q}(X_{J,\gamma}(0), \mathbb{C}).$ The proof is completed.
\end{proof}

We have the following result concerning cohomology of contact loci.

\begin{coro}\label{corconstantsheaf}
Let $f\in\mathbb C[x_1,\dots,x_d]$ be nondegenerate in the sense of Kouchnirenko, $n\in \mathbb N^*$ and $f(O)=0$. Then, there is a spectral sequence
\begin{equation*}
E^{p,q}_1\Longrightarrow H^{p+q}_c(\mathscr X_{n,O}(f), \mathbb{C}),
\end{equation*}
where
\[
E^{p,q}_1=\bigoplus _{\gamma\in K}\bigoplus_{J\supseteq J_{\gamma}}\Big(H_{p-q}(X_{J,\gamma}(1), \mathbb{C})^{|D_{J,\gamma,0,p}^{(n)}|}
\oplus\bigoplus_{k\geq 1}H_{p-q}(X_{J,\gamma}(0), \mathbb{C})^{|D_{J,\gamma,k,p}^{(n)}|}\Big).
\]
\end{coro}

\begin{proof}
Apply Theorem~\ref{thm5.6} for $\mathcal{F}$ to be the constant sheaf on $\mathscr X_{n,O}(f)$ associated to the field of complex numbers $\mathbb{C}$, since the inverse image of constant sheaf is a constant sheaf, the Corollary is a direct consequence of Theorem~\ref{thm5.6} and Lemmas~\ref{lem46} and~\ref{lem47}.
\end{proof}

\subsection*{Acknowledgements}
The first author is deeply grateful to Fran\c cois Loeser for introducing him to Problem~\ref{pro1}.
%His research is supported by the Tosio Kato Fellowship awarded in 2019.
He thanks Department of Mathematics, Graduate School of Science, Osaka University for warm hospitality during his visit as a  Specially Appointed Researcher supported by the MSJ Tosio Kato Fellowship.
The authors thank Vietnam Institute for Advanced Study in Mathematics (VIASM) and Department of Mathematics - KU Leuven for warm hospitality during their visits.

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