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

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 ${\mathrm{deg}}_{Y}f\le 2$, 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.