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

[摘要] In [6], Murray Marshall proved that every f epsilon R[X, Y] non-negative on the strip [0, 1] x R can be written as f = sigma(0) + sigma X-1(1 - X) with sigma(0), sigma(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] x 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 partial derivative f/partial derivative X does not vanish at any of them. (C) 2020 Elsevier B.V. All rights reserved.

[发布日期] 2021-07-01

[效力级别]  Proceedings Paper

[关键词] Certificates of non-negativity;Sums of squares;Degree bounds