Logarithmic Hesse's Problem

TL;DR

This paper proves that Laurent polynomials with vanishing logarithmic Hessian are essentially functions of fewer variables, up to automorphism, revealing a structural property related to Hesse's problem.

Contribution

It establishes a new connection between vanishing logarithmic Hessian and variable dependence for Laurent polynomials on complex tori.

Findings

Laurent polynomials with zero logarithmic Hessian depend on fewer variables

Up to automorphism, such polynomials are reducible to functions of at most n-1 variables

Provides insight into the structure of polynomials with vanishing logarithmic Hessian

Abstract

We show that if a Laurent polynomial on the coordinate ring of the complex algebraic torus on n variables has vanishing logarithmic Hessian, then up to an automorphism of the torus, the Laurent polynomial depends on at most n-1 variables.