A refinement of the Kushnirenko-Bernstein estimate

Discrete Mixed Volume and Hodge-Deligne Numbers

Generalizing the famous Bernstein-Kushnirenko Theorem, Khovanskii proved in 1978 a combinatorial formula for the arithmetic genus of the compactification of a generic complete intersection associated to a family of lattice polytopes. Recently, an analogous combinatorial formula, called the discrete mixed volume, was introduced by Bihan and shown to be nonnegative. By making a footnote of Khovanskii in his paper explicit, we interpret this invariant as the (motivic) arithmetic genus of the non-compact generic complete intersection associated to the family of lattice polytopes.

Algebraic equations and convex bodies

The well-known Bernstein-Kushnirenko theorem from the theory of Newton polyhedra relates algebraic geometry and the theory of mixed volumes. Recently the authors have found a far-reaching generalization of this theorem to generic systems of algebraic equations on any quasi-projective variety. In the present note we review these results and their applications to algebraic geometry and convex geometry.

二维离散周期薛定谔算子谱带函数的极值

Extrema of spectral band functions of two dimensional discrete periodic Schrödinger operators

本文利用贝祖定理和BKK定理分析二维离散周期薛定谔算子谱带函数极值的水平集，改进了先前结果。

How many zeroes? Counting the number of solutions of systems of polynomials via geometry at infinity (Draft III)

In this book we describe an approach through toric geometry to the following problem: "estimate the number (counted with appropriate multiplicity) of isolated solutions of n polynomial equations in n variables over an algebraically closed field k." The outcome of this approach is the number of solutions for "generic" systems in terms of their "Newton polytopes," and an explicit characterization of what makes a system "generic." The pioneering work in this field was done in the 1970s by Kushnirenko, Bernstein and Khovanskii, who completely solved the problem of counting solutions of generic sys…

Intersection multiplicity, Milnor number and Bernstein's theorem

We explicitly characterize when the Milnor number at the origin of a polynomial or power series (over an algebraically closed field k of arbitrary characteristic) is the minimum of all polynomials with the same Newton diagram, which completes works of Kushnirenko (Invent. Math., 1976) and Wall (J. Reine Angew. Math., 1999). Given a fixed collection of n convex integral polytopes in R^n, we also give an explicit characterization of systems of n polynomials supported at these polytopes which have the maximum number (counted with multiplicity) of isolated zeroes on k^n, or more generally, on a un…

Multiplicity estimates, analytic cycles and Newton polytopes

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 particular we obtain simpler proofs to many of the best known estimates, and give more general formulations in terms of New…