Nonnegative Polynomial with no Certificate of Nonnegativity in the Simplicial Bernstein Basis

TL;DR

This paper constructs a specific nonnegative polynomial that defies representation with nonnegative coefficients in the simplicial Bernstein basis, highlighting limitations of Bernstein's theorem in certifying nonnegativity.

Contribution

It provides a counterexample demonstrating that Bernstein's theorem does not extend to certificates of nonnegativity for polynomials with isolated zeros.

Findings

Counterexample of a nonnegative polynomial with no Bernstein basis certificate

Shows limitations of Bernstein's theorem in polynomial nonnegativity certification

Highlights need for alternative methods in polynomial nonnegativity proofs

Abstract

This paper presents a nonnegative polynomial that cannot be represented with nonnegative coefficients in the simplicial Bernstein basis by subdividing the standard simplex. The example shows that Bernstein Theorem cannot be extended to certificates of nonnegativity for polynomials with zeros at isolated points.