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

Abstract

In [6] , Murray Marshall proved that every f ∈ R [ X , Y ] non-negative on the strip [ 0 , 1 ] × R can be written as f = σ 0 + σ 1 X ( 1 − X ) with σ 0 , σ 1 sums of squares in R [ X , Y ] . In this work, we present a few results concerning this representation in particular cases. First, under the assumption deg Y ⁡ f ≤ 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 ] × 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 ∂ f ∂ X does not vanish at any of them.