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

Abstract

In Polynomials non-negative on a strip, Murray Marshall proved that every non-negative on the strip can be written as with sums of squares in . In this work, we present a few results concerning this representation in particular cases. First, under the assumption , 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 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 does not vanish at any of them.