A remark on the approximation of non-negative polynomials by SONC polynomials

TL;DR

This paper demonstrates that certain non-negative polynomials cannot be approximated arbitrarily closely by SONC polynomials, highlighting limitations in the approximation capabilities of SONC methods.

Contribution

It establishes the existence of non-negative polynomials that cannot be uniformly approximated by SONC polynomials, revealing fundamental limitations of SONC-based approximation.

Findings

Some non-negative polynomials are not uniformly approximable by SONC polynomials.

SONC polynomials have inherent approximation limitations.

The paper clarifies the scope of SONC polynomial approximations.

Abstract

A SONC polynomial is a sum of finitely many non-negative circuit polynomials, whereas a non-negative circuit polynomial is a non-negative polynomial whose support is a simplicial circuit. We show that there exist non-negative polynomials that cannot be uniformly approximated by SONC polynomials arbitrarily well.