Chennai Mathematical Institute


3:30 pm, Seminar Hall
Counting zeroes of Laurent polynomials in the algebraic torus

Jugal Verma


Bernstein proved in 1975 that the number of zeroes in $\mathbb (C^{*})^n$ of $n$ Laurent polynomials $f_1,f_2,\ldots,f_n$ in $n$ indeterminates is bounded above by the mixed volumes of their Newton polytopes. This Theorem while generalising classical theorem of Bezout, provides better bounds for number of common zeros of sparse polynomials.

There are several proofs of this theorem using diverse techniques such as homotopy continuation methods in numerical analysis, theory of mixed volumes of polytopes in convex geometry, Riemann-Roch theorem for toric varieties and Hilbert functions of multigraded algebras. We will outline a purely algebraic proof which connects mixed multiplicities of ideals with mixed volumes of polytopes.