Toric extensions of P\'olya's theorem

TL;DR

This paper extends Pólya's theorem to sparse polynomials using toric geometry, improving copositivity certification and applying it to analyze Feynman integrals in physics.

Contribution

It introduces new extensions and converses to Pólya's theorem for sparse polynomials via toric geometry, and explores applications to Feynman integral convergence.

Findings

Extended Pólya's theorem for sparse polynomials.

Provided conditions for copositivity detection.

Applied methods to Feynman integral analysis.

Abstract

The classical version of P\'olya's theorem provides a simple method for certifying that a homogeneous polynomial of degree d is strictly copositive, that is, it takes only positive values on the nonnegative real orthant. However, this method might fail to detect copositivity of polynomials that are missing certain degree d monomials. In this paper, we present extensions and converses to P\'olya's theorem for sparse polynomials, using techniques from positive toric geometry. Furthermore, we explore how this method can be used to study the convergence of Feynman integrals in particle physics.