Stubborn Polynomials

TL;DR

This paper characterizes stubborn polynomials, which are nonnegative on real varieties but whose odd powers are not sums of squares, revealing their existence depends on the geometry and degree of the variety.

Contribution

It provides a complete characterization of stubborn polynomials on smooth curves and proves a conjecture relating stubbornness to the real delta-invariant for ternary sextics.

Findings

Stubborn polynomials on smooth totally real curves are characterized by all zeros being real.

Existence of stubborn polynomials depends on the genus and degree of the curve.

Confirmed the conjecture that nonnegative ternary sextics are stubborn iff their real delta-invariant is at least 9.

Abstract

The relationship between nonnegative polynomials and sums of squares is a classical topic in real algebraic geometry. We study \emph{stubborn polynomials} $f$ on a real variety $X$, which are polynomials nonnegative on $X$, such that no odd power of $f$ is a sum of squares. Previously, stubborn polynomials were studied only in the globally nonnegative case, with results restricted to polynomials nonnegative on $\mathbb{P}^2$. We fully characterize stubborn polynomials on smooth curves, showing that a polynomial on a smooth totally real curve is stubborn if and only if all of its zeros are real. This implies that there exist smooth curves with no stubborn polynomials in low degree, while stubborn polynomials must exist in sufficiently high degrees on curves with positive genus. We explore the much more delicate situation with singular and reducible curves. While being real-rooted always implies being stubborn, there also exist singular curves with no stubborn polynomials at all. We then analyze the case of ternary sextics, i.e.,~polynomials of degree $6$ on $\mathbb{P}^2$. We prove the Conjecture of Blekherman, Kozhasov, and Reznick that a nonnegative ternary sextic is stubborn if and only if its real delta-invariant is at least 9. To analyze this case, we develop results for lifting stubborn polynomials from curves to higher dimensional varieties and use the theory of weak Del Pezzo surfaces. We complement these results with structural properties of stubborn polynomials and present many explicit examples.