Intersection multiplicity, Milnor number and Bernstein's theorem

This paper is a survey on major results on Hilbert functions of multigraded algebras and mixed multiplicities of ideals, including their applications to the computation of Milnor numbers of complex analytic hypersurfaces with isolated singularity, multiplicities of blowup algebras and mixed volumes of polytopes.

Using invariants from commutative algebra to count geometric objects is a basic idea in singularities. For example, the multiplicity of an ideal is used to count points of intersection of two analytic sets at points of non-transverse intersection. A problem with the multiplicity of an ideal or module is that it is only defined for modules and ideals of finite colength. In this paper we use pairs of modules and their multiplicities as a way around this difficulty. A key tool...

We present a quite efficient method to compute the roots of Bernstein-Sato polynomial of a homogeneous polynomial if the associated projective hypersurface has only weighted homogeneous isolated singularities (so that its local Bernstein-Sato polynomials are uniquely determined by weights) and if a certain condition is satisfied. In the three variable case, the last condition holds except for polynomials of quite special type (that is, extremely degenerated ones) as far as ca...

We consider the problem of estimating the multiplicity of a polynomial when restricted to the smooth analytic trajectory of a (possibly singular) polynomial vector field at a given point or points, under an assumption known as the D-property. Nesterenko has developed an elimination theoretic approach to this problem which has been widely used in transcendental number theory. We propose an alternative approach to this problem based on more local analytic considerations. In p...

In this note we determine the Bernstein-Sato polynomial $b_Q(s)$ of a generic central arrangement $Q=\prod_{i=1}^kH_i$ of hyperplanes. We establish a connection between the roots of $b_Q(s)$ and the degrees of the generators for the top cohomology of the corresponding Milnor fiber. This connection holds for all homogeneous polynomials. We also introduce certain subschemes of the arrangement determined by the roots of $b_Q(s)$.

A zero-dimensional polynomial ideal may have a lot of complex zeros. But sometimes, only some of them are needed. In this paper, for a zero-dimensional ideal $I$, we study its complex zeros that locate in another variety $\textbf{V}(J)$ where $J$ is an arbitrary ideal. The main problem is that for a point in $\textbf{V}(I) \cap \textbf{V}(J)=\textbf{V}(I+J)$, its multiplicities w.r.t. $I$ and $I+J$ may be different. Therefore, we cannot get the multiplicity of this point w....

In this paper we prove the Gromov--Milman conjecture (the Dvoretzky type theorem) for homogeneous polynomials on $\mathbb R^n$, and improve bounds on the number $n(d,k)$ in the analogous conjecture for odd degrees $d$ (this case is known as the Birch theorem) and complex polynomials. We also consider a stronger conjecture on the homogeneous polynomial fields in the canonical bundle over real and complex Grassmannians. This conjecture is much stronger and false in general, but...

We give an elementary introduction to some recent polyhedral techniques for understanding and solving systems of multivariate polynomial equations. We provide numerous concrete examples and illustrations, and assume no background in algebraic geometry or convex geometry. Highlights include the following: (1) A completely self-contained proof of an extension of Bernstein's Theorem. Our extension relates volumes of polytopes with the number of connected components of the comp...

Geometry of sparse systems of polynomial equations (i.e. the ones with prescribed monomials and generic coefficients) is well studied in terms of their Newton polytopes. The results of this study are colloquially known as the Bernstein--Kouchnirenko--Khovanskii toolkit, and unfortunately are not applicable to many important systems, whose coefficients slightly fail to be generic. This for instance happens if some of the equations are obtained from another one by taking part...

The Milnor number of an isolated hypersurface singularity, defined as the codimension $\mu(f)$ of the ideal generated by the partial derivatives of a power series $f$ whose zeros represent locally the hypersurface, is an important topological invariant of the singularity over the complex numbers, but its meaning changes dramatically when the base field is arbitrary. It turns out that if the ground field is of positive characteristic, this number is not even invariant under co...