Aversion of Putinar's Positivstellensatz for cylinders

[摘要] We prove that, under some additional assumption, Putinar's Positivstellensatz holds on cylinders of type S x R with S={(x) over bar is an element of R-n vertical bar g(1)((x) over bar) >= 0,..., g(s)((x) over bar) >= 0} such that the quadratic module generated by g(1),..., g(s) in R[X-1 ,..., X-n] is archimedean, and we provide a degree bound for the representation of a polynomial f is an element of R[X-1,..., X-n, Y] which is positive on S x R as an explicit element of the quadratic module generated by g(1),..., g(s) in R [ X-1 ,..., X-n, Y]. We also include an example to show that an additional assumption is necessary for Putinar's Positivstellensatz to hold on cylinders of this type. (C) 2020 Elsevier B.V. All rights reserved.

[发布日期] 2020-12-01

[关键词] Putinar's Positivstellensatz;Sums of squares;Degree bounds