Polynomials nonnegative on the cylinder

TL;DR

This paper extends the understanding of nonnegative polynomials on certain real algebraic curves, proving they are sums of squares under specific conditions, with a complete result for the circle.

Contribution

It generalizes Marshall's strip conjecture to affine nonsingular curves with compact real points, establishing sum of squares representations for nonnegative polynomials.

Findings

Affirmative answer for polynomials with finitely many zeros.

Complete proof for the circle case.

Extension of sum of squares results to broader classes of curves.

Abstract

In 2010, Marshall settled the strip conjecture, according to which every polynomial in $\mathbb{R}[x,y]$, nonnegative on the strip $[-1,1]\times\mathbb{R}$, is a sum of squares and of squares times $1-x^2$. We consider affine nonsingular curves $C$ over $\mathbb{R}$ with $C(\mathbb{R})$ compact, and study the question whether every $f$ in $\mathbb{R}[C][y]$, nonnegative on $C(\mathbb{R})\times\mathbb{R}$, is a sum of squares in $\mathbb{R}[C][y]$. We give an affirmative answer under the condition that $f$ has only finitely many zeros in $C(\mathbb{R})\times\mathbb{R}$. For $C$ the circle $x_1^2+x_2^2=1$, we prove the result unconditionally.