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

PAULA ESCORCIELO; DANIEL PERRUCCI

JOURNAL OF PURE AND APPLIED ALGEBRA

ELSEVIER SCIENCE BV

Lugar: Amsterdam; Año: 2021 vol. 225

0022-4049

Murray Marshall proved that every f∈R[X,Y] non-negative on the strip [0,1]×R can be written as f=σ0+σ1X(1−X) with σ0,σ1 sums of squares in R[X,Y]. In this work, wepresent a few results concerning this representation in particular cases. First, under the assumption degYf≤2, by characterizing the extreme rays of a suitable cone, we obtain a degree bound foreach term. Then, we consider the case offpositive 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 thesum 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.