Alladi Ramakrishnan Hall

Counting Zeros of  Multivariate Laurent Polynomials and Mixed Volumes of Polytopes

Jugal Verma

IIT Bombay

A result of D.N. Bernstein proved in the late seventies gives an upper bound on the number of common solutions of n multivariate Laurent polynomials in n indeterminates in terms of  the mixed volumes of their Newton  polytopes. This bound refines the classical Bezout's bound. Bernstein's Theorem  has several proofs using techniques from numerical analysis, intersection theory and toric varieties. B. Teissier proved the theorem using intersection theory. A proof using theory of toric varieties can be found in the book by W. Fulton on the same subject.

In this talk, I will outline an algebraic proof similar to the standard proof of Bezout's Theorem. This proof, found in collaboration with N. V. Trung, uses  basic  results  about  Hilbert functions of multigraded algebras first proved by van der Waerden.