On sum of squares certificates of non-negativity on a strip

TL;DR

This paper investigates sum of squares certificates for non-negative polynomials on a strip, providing degree bounds and constructive methods for specific cases, including bounded degree in one variable and boundary zeros.

Contribution

It offers new degree bounds and constructive techniques for sum of squares representations of non-negative polynomials on a strip under various conditions.

Findings

Degree bounds for polynomials with degree in Y at most 2.

Constructive method for positive polynomials non-vanishing at infinity.

Method applicable to polynomials with boundary zeros where derivatives do not vanish.

Abstract

A well-known result of Murray Marshall states that every $f \in \mathbb{R} [X,Y]$ non-negative on the strip $[0,1] \times \mathbb{R}$ can be written as $f= \sigma_0 + \sigma_1 X(1-X)$ with $\sigma_0, \sigma_1$ sums of squares in $\mathbb{R} [X,Y]$. In this work, we present a few results concerning this representation in particular cases. First, under the assumption ${\rm deg}_Y f \leq 2$, by characterizing the extreme rays of a suitable cone, we obtain a degree bound for each term. Then, we consider the case of $f$ positive on $[0,1] \times \mathbb{R}$ and non-vanishing at infinity, and we show again a degree bound for each term, coming from a constructive method to obtain the sum of squares representation. Finally, we show that this constructive method also works in the case of $f$ having only a finite number of zeros, all of them lying on the boundary of the strip, and such that $\frac{\partial f}{\partial X}$ does not vanish at any of them.