Work

On odd powers of nonnegative polynomials that are not sums of squares

Year: 2026

Type: article

Abstract: We initiate a systematic study of nonnegative polynomials P such that $P^k$ is not a sum of squares for any odd $k\geq 1$ , calling such P stubborn . We develop a new invariant of a real isolated zero of a nonnegative polynomial in the plane, that we call the SOS-invariant , and relate it to the well-known delta invariant of a plane curve singularity. Using the SOS-invariant we show that any polynomial that spans an extreme ray of the convex cone of nonnegative ternary forms of degree 6 is stubborn. We also show how to use the SOS-invariant to prove stubbornness of ternary forms in higher degree. Furthermore, we prove that in a given degree and number of variables, nonnegative polynomials that are not stubborn form a convex cone, whose interior consists of all strictly positive polynomials.

Source: Forum of Mathematics Sigma

DOI: 10.1017/fms.2026.10221

Authors: Grigoriy Blekherman, Khazhgali Kozhasov, Bruce Reznick

Institutions: Georgia Institute of Technology, Université Côte d'Azur, Communauté d'universités et d'établissements Université Côte d'Azur, University of Illinois Urbana-Champaign, University of Illinois System

Language: English

Topic: Advanced Optimization Algorithms Research

Subfield: Numerical Analysis

Field: Mathematics

Domain: Physical Sciences

Open Access status: gold

Estimated APC paid: $2,650

OpenAlex ID: W4402733595