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

A well-known result of Murray Marshall states that every $f \in $\mathbb{R}$ [X,Y]$non-negativeonthestrip$[0,1] \times $\mathbb{R}$$canbewrittenas$f= \sigma_0 + \sigma_1 X(1-X)$with$\sigma_0, \sigma_1$sumsofsquaresin$$\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.