Lê numbers and Newton diagram

Let $(X,\mathscr{O}_{X})$ be a complex analytic space, $W\subseteq X$ be an analytic subset of $X$, and $\mathscr{I}$ be a coherent sheaf of ideals in $\mathscr{O}_{X}$. As usual, we denote by $V(\mathscr{I})$ the analytic space defined by the vanishing of $\mathscr{I}$. At each point $x\in V(\mathscr{I})$, we want to consider scheme-theoretically those components of $V(\mathscr{I})$ which are not contained in $W$. For this purpose, we look at a minimal primary decomposition of the stalk $\mathscr{I}_{x}$ of $\mathscr{I}$ in the local ring $\mathscr{O}_{X,x}$, and we consider the ideal $\mathscr{I}_{x}\lnot W$ in $\mathscr{O}_{X,x}$ consisting of the intersection of those (possibly embedded) primary components $Q$ of $\mathscr{I}_{x}$ such that $V(Q)\nsubseteq W$. This definition does not depend on the choice of the minimal primary decomposition of $\mathscr{I}_{x}$. Now, if we perform the operation described above at the point $x$ simultaneously at all points of $V(\mathscr{I})$, then we obtain a coherent sheaf of ideals called a gap sheaf and denoted by $\mathscr{I}\lnot W$. Hereafter, we shall denote by $V(\mathscr{I})\lnot W$ the scheme (i.e., the complex analytic space) $V(\mathscr{I}\lnot W)$ defined by the vanishing of the gap sheaf $\mathscr{I}\lnot W$.

Consider an analytic function $f\colon(U,0)\rightarrow(\mathbb{C},0)$, where $U$ is an open neighbourhood of $0$ in $\mathbb{C}^{n}$, and fix a system of linear coordinates $z=(z_{1},\ldots,z_{n})$ for $\mathbb{C}^{n}$. Let $\Sigma f$ be the critical locus of $f$. For $0\leq k\leq n-1$, the $k$th (relative) polar variety of $f$ with respect to the coordinates $z$ is the scheme

The analytic cycle

is called the $k$th Lê cycle of $f$ with respect to the coordinates $z$. (We always use brackets $[\cdot]$ to denote analytic cycles.) The $k$th Lê number $\lambda^{k}_{f,z}(0)$ of $f$ at $0\in\mathbb{C}^{n}$ with respect to the coordinates $z$ is defined to be the intersection number

provided that this intersection is $0$-dimensional or empty at $0$; otherwise, we say that $\lambda^{k}_{f,z}(0)$ is undefined.¹¹1As usual, $[V(z_{1},\ldots,z_{k})]$ denotes the analytic cycle associated to the analytic space defined by $z_{1}=\cdots=z_{k}=0$. The notation $\big{(}[\Lambda^{k}_{f,z}]\cdot[V(z_{1},\ldots,z_{k})]\big{)}_{0}$ stands for the intersection number at $0$ of the analytic cycles $[\Lambda^{k}_{f,z}]$ and $[V(z_{1},\ldots,z_{k})]$. For $k=0$, the relation (2.1) means

For any $\dim_{0}\Sigma f<k\leq n-1$, the Lê number $\lambda^{k}_{f,z}(0)$ is always defined and equal to zero. For this reason, we usually only consider the Lê numbers

Note that if $0$ is an isolated singularity of $f$, then $\lambda^{0}_{f,z}(0)$ (which is the only possible non-zero Lê number) is equal to the Milnor number $\mu_{f}(0)$ of $f$ at $0$.

As already mentioned above, a key ingredient in the proof of our main result is the Iomdine-Lê-Massey formula (see [12, Theorem 4.5]). Roughly, this formula says that if the Lê numbers of $f$ at $0$ with respect to $z$ exist and if $d:=\dim_{0}\Sigma f\geq 1$, then for any integer $\alpha_{1}$ large enough, $\dim_{0}\Sigma(f+z_{1}^{\alpha_{1}})=d-1$ and the Lê numbers of $f+z_{1}^{\alpha_{1}}$ at $0$ with respect to the rotated coordinates $z^{(1)}:=(z_{2},\ldots,z_{n},z_{1})$ exist and they can be described in terms of the Lê numbers $\lambda_{f,z}^{k}(0)$ of the original function $f$. Moreover, the formula says that any $\alpha_{1}>\rho_{f,z}(0)$ is suitable, where $\rho_{f,z}(0)$ is the maximum “polar ratio” of $f$ at $0$ with respect to $z$. In this section, we recall the definition of polar ratios (see [12, Definition 4.1]).

The notation is as in Section 2.2. Suppose that $\dim_{0}\Gamma^{1}_{f,z}=1$. Let $\eta$ be an irreducible component of $\Gamma^{1}_{f,z}$ (with its reduced structure) such that $\dim_{0}(\eta\cap V(z_{1}))=0$. The polar ratio of $\eta$ at $0$ is the number defined by

If $\dim_{0}(\eta\cap V(z_{1}))\not=0$, then we say that the polar ratio of $\eta$ at $0$ is equal to $1$. A polar ratio for $f$ at $0$ with respect to $z$ is any one of the polar ratios at $0$ of any component of $\Gamma^{1}_{f,z}$.

For example, if $f$ is a homogeneous polynomial and if $\dim_{0}\Gamma^{1}_{f,z}=1$, then each component of $\Gamma^{1}_{f,z}$ is a line, and hence the polar ratios for $f$ at $0$ with respect to $z$ are all equal to $1$ or to the degree $\deg(f)$ of the polynomial $f$ (see [12, Remark 4.2]).

In [13, Section 3.2], M. Morgado and M. Saia gave an upper bound for the maximal polar ratio for a semi-weighted homogeneous arrangement.

Here, the reference is Kouchnirenko [7]. The Newton polyhedron $\Gamma_{\!+}(f)$ of $f$ (at the origin and with respect to the coordinates $z=(z_{1},\ldots,z_{n})$) is the convex hull in $\mathbb{R}_{+}^{n}$ of the set

For any $v\in\mathbb{R}_{+}^{n}\setminus\{0\}$, put

where $\langle\cdot\,,\cdot\rangle$ denotes the standard inner product in $\mathbb{R}^{n}$. A subset $\Delta\subseteq\Gamma_{\!+}(f)$ is called a face of $\Gamma_{\!+}(f)$ if there exists $v\in\mathbb{R}_{+}^{n}\setminus\{0\}$ such that $\Delta=\Delta(v,\Gamma_{\!+}(f))$. The dimension of a face $\Delta$ of $\Gamma_{\!+}(f)$ is the minimum of the dimensions of the affine subspaces of $\mathbb{R}^{n}$ containing $\Delta$. The Newton diagram (also called Newton boundary) of $f$ is the union of the compact faces of $\Gamma_{\!+}(f)$. It is denoted by $\Gamma(f)$. We say that $f$ is convenient if the intersection of $\Gamma(f)$ with each coordinate axis of $\mathbb{R}^{n}_{+}$ is non-empty (i.e., for any $1\leq i\leq n$, the monomial $z_{i}^{\alpha_{i}}$, $\alpha_{i}\geq 1$, appears in the expression $\sum_{\alpha}c_{\alpha}z^{\alpha}$ with a non-zero coefficient).

For any face $\Delta\subseteq\Gamma(f)$, define the face function $f_{\Delta}$ by

We say that $f$ is Newton non-degenerate (in short, non-degenerate) on the face $\Delta$ if the equations

have no common solution on $(\mathbf{\mathbb{C}}\setminus\{0\})^{n}$. We say that $f$ is (Newton) non-degenerate if it is non-degenerate on every face $\Delta$ of $\Gamma(f)$.

Another important ingredient in the proof of our main theorem is Lemma 3.7 of [1]. This lemma asserts that if $f$ is a non-degenerate function with a singularity at $0$, then there exists a constant $m(f)$ such that for any $\alpha_{i}>m(f)$, the function $f+z_{i}^{\alpha_{i}}$ is non-degenerate too. Such a (non unique) number $m(f)$ is defined as follows. For each face $\Delta\subseteq\Gamma(f)$ with maximal dimension (i.e., $\Delta$ is not contained in any other face), choose a vector $v_{\Delta}\in\mathbb{R}_{+}^{n}\setminus\{0\}$ such that $\Delta=\{\alpha\in\Gamma_{\!+}(f)\,;\,\langle v_{\Delta},\alpha\rangle=\ell(v_{\Delta},\Gamma_{\!+}(f))\}$, and define

where the union is taken over all maximal dimensional faces $\Delta\subseteq\Gamma(f)$. Clearly, $W$ is a compact set and it intersects each coordinate axis of $\mathbb{R}_{+}^{n}$ in a closed interval, say $[0,w_{i}]$ for some $w_{i}$. Then define

Of course, $m(f)$ depends on the choice of the vectors $v_{\Delta}$. It is possible to define a “smallest” number $m_{0}(f)$ that also guarantees the non-degeneracy of the functions $f+z_{i}^{\alpha_{i}}$ with $\alpha_{i}>m_{0}(f)$ (see [6, Section 2]). However, for our purpose, we shall not need it.

Again, the reference for this section is [7]. Throughout the paper, for any subsets $I\subseteq\{1,\ldots,n\}$ and $X\subseteq\mathbb{R}^{n}_{+}$, we shall use the following notation:

In particular, for any $i\in\{1,\ldots,n\}$, the set $X^{\{i\}}$ is nothing but the intersection of $X$ with the $i$th coordinate axis of $\mathbb{R}^{n}_{+}$.

Let $\Gamma_{\!-}(f)$ denote the cone over $\Gamma(f)$ with the origin as vertex. If $f$ is convenient, then the Newton number $\nu(f)$ of $f$ is defined by

where $|I|$ is the cardinality of $I$ and $\operatorname{Vol}_{|I|}(\Gamma_{\!-}(f)^{I})$ is the $|I|$-dimensional Euclidean volume of $\Gamma_{\!-}(f)^{I}$. For $I=\emptyset$, the subset $\Gamma_{\!-}(f)^{\emptyset}$ reduces to $\{0\}$, and we set $\operatorname{Vol}_{0}(\Gamma_{\!-}(f)^{\emptyset})=1$.

The Newton number can also be defined even if $f$ is not convenient. More precisely, if $I$ is the non-empty subset of $\{1,\ldots,n\}$ such that $\Gamma(f)$ meets the $i$th coordinate axis of $\mathbb{R}^{n}_{+}$ if and only if $i\notin I$, then the Newton number $\nu(f)$ of $f$ is defined as

where of course the Newton number of the (convenient) function $f+\sum_{i\in I}z_{i}^{m}$ is given by (3.1).

Following Kouchnirenko’s definition of the Newton number, we now introduce our modified Newton numbers.

Let $I$ be a non-empty subset of $\{1,\ldots,n\}$ such that $\Gamma(f)^{I}\not=\emptyset$. By [3, Theorem 1], choose a simplicial decomposition of $\Gamma(f)^{I}$ in which the vertices of a simplex are $0$-dimensional faces of $\Gamma(f)^{I}$ (such a decomposition is not unique). The cones spanned by the origin $0\in\mathbb{R}^{n}$ and such simplexes give a simplicial decomposition

of $\Gamma_{\!-}(f)^{I}$. Note that

Clearly, each simplex $S_{I,r}\subseteq(\mathbb{R}^{n}_{+})^{I}$ may be identified to a simplex (still denoted by $S_{I,r}$) of $\mathbb{R}^{|I|}$, and with such an identification, the volume $\operatorname{Vol}_{|I|}(S_{I,r})$ of a simplex $S_{I,r}$ with maximal dimension (i.e., with dimension $|I|$) is given by

where $0,S_{I,r;1},\ldots,S_{I,r;|I|}$ are the column vectors representing the coordinates of the vertices of the simplex $S_{I,r}\subseteq\mathbb{R}^{|I|}$. Note that each such column vector has $|I|$ components, so that the matrix in (3.3) has dimension $(|I|+1)\times(|I|+1)$.

Let $J$ be another subset of $\{1,\ldots,n\}$. We suppose that for any $i\in J$ the Newton boundary $\Gamma(f)$ meets the $i$th coordinate axis of $\mathbb{R}^{n}_{+}$. Then to each $i_{0}\in\{1,\ldots,n\}$, we associate a subset $\Xi_{I,J,i_{0}}$ of $\Xi_{I}$ (depending on $I$, $J$ and $i_{0}$) as follows. If $i_{0}\in I\cap J$, then we define $\Xi_{I,J,i_{0}}$ as the set of all simplexes $S_{I,r}\in\Xi_{I}$ (as simplexes in $(\mathbb{R}^{n}_{+})^{I}$) with maximal dimension $|I|$ such that for any $i\in J$ the following property holds true:

(As usual, by an “edge” of a simplex we mean a $1$-dimensional face.) If $i_{0}\notin J$ (in particular if $J=\emptyset$) or if $i_{0}\notin I$, then we set $\Xi_{I,J,i_{0}}:=\emptyset$.

By definition, if $S_{I,r}$ is a simplex of $\Xi_{I,J,i_{0}}$, then it has maximal dimension and possesses a vertex with coordinates of the form $(0,\ldots,0,\alpha_{i_{0}},0,\ldots,0)\in\mathbb{R}^{n}$ (for some $\alpha_{i_{0}}$ located at the $i_{0}$th place). To each such a simplex $S_{I,r}\in\Xi_{I,J,i_{0}}$, we associate a (unique) “reduced” simplex $\widetilde{S}_{I,r}$ defined by the same vertices as those of $S_{I,r}$ with the exception of the vertex $(0,\ldots,0,\alpha_{i_{0}},0,\ldots,0)$ which we replace by $(0,\ldots,0,1,0,\ldots,0)$. We denote by $\widetilde{\Xi}_{I,J,i_{0}}$ the set of such reduced simplexes.

By convention, for the next definition and all the statements hereafter, we agree that if $I$ is a non-empty subset of $\{1,\ldots,n\}$ such that $\Gamma(f)^{I}$ is empty, then the corresponding “simplicial decomposition” $\Xi_{I}$ is the empty set.

Similarly, we introduce the subset $\Xi_{I,J,0}$ of $\Xi_{I}$ consisting of those simplexes $S_{I,r}\in\Xi_{I}$ with maximal dimension and such that for any $i\in J$ the intersection $S_{I,r}^{\{i\}}=S_{I,r}\cap\Gamma_{\!-}(f)^{\{i\}}$ is not an edge of $S_{I,r}$.

Let us emphasize the fact that the simplexes involved in the definition of the modified Newton number $\widetilde{\nu}_{\Xi,J,i_{0}}(f)$ are reduced simplexes, while those used to define the special modified Newton number ${\nu}_{\Xi,J,0}(f)$ are not reduced.

We need to compute the polar varieties $\Gamma^{2}_{f,z}$ and $\Gamma^{1}_{f,z}$ and the Lê cycles $[\Lambda^{1}_{f,z}]$ and $[\Lambda^{0}_{f,z}]$. By definition,

while

It follows that the Lê cycles are given by

and