Representation of nonnegative convex polynomials

TL;DR

This paper characterizes nonnegative convex polynomials on certain semi-algebraic sets, showing they can be represented within a specific quadratic module generated by defining polynomials.

Contribution

It introduces a new representation for nonnegative convex polynomials on semi-algebraic sets, expanding understanding of polynomial positivity in non-compact domains.

Findings

Nonnegative convex polynomials belong to a specific quadratic module.

Representation applies to convex, not necessarily compact, semi-algebraic sets.

Provides a constructive way to verify polynomial nonnegativity.

Abstract

We provide a specific representation of convex polynomials nonnegative on a convex (not necessarily compact) basic closed semi-algebraic subset K of Rn. Namely, they belong to a specific subset of the quadratic module generated by the concave polynomials that define K.