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Article · Forum of Mathematics Sigma · January 1, 2026

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

Grigoriy Blekherman, Khazhgali Kozhasov, Bruce A. Reznick

Khazhgali Kozhasov
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Abstract

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.

Advanced Optimization Algorithms ResearchAlgebraic Geometry and Number TheoryPolynomial and algebraic computationnonnegative polynomialssums of squaresdelta-invariant

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