Positivity of polynomials on the nonnegative part of certain affine hypersurfaces

TL;DR

This paper extends Pólya's positivity result to polynomials positive on specific affine hypersurfaces, showing they can be represented with positive coefficients using real algebra techniques.

Contribution

It generalizes Pólya's theorem to a broader class of semi-algebraic sets defined by affine hypersurfaces with positive coefficients.

Findings

Polynomials positive on the set can be expressed with positive coefficients

The result generalizes Pólya's theorem for the standard simplex

Uses Archimedean Representation Theorem in the proof

Abstract

We consider polynomials on the intersection of the closed positive orthant with the height-$1$ level hypersurface of certain polynomials with positive coefficients. We show that any polynomial strictly positive on such a semi-algebraic set can be represented by some polynomial with only positive coefficients. This result generalizes a result of P\'olya which corresponds to the case when the semi-algebraic set is the standard simplex. Our proof uses the Archimedean Representation Theorem from real algebra.