A version of Putinar's Positivstellensatz for cylinders

DANIEL PERRUCCI; PAULA ESCORCIELO

Madrid

Workshop; Effective Methods in Real Algebraic Geometry (MEGAR) 2019; 2019

Universidad Complutense de Madrid

We prove that, under some additional assumption, Putinar?s Positivstellensatz holds on cylinders of type S × R with S = {x̄ ∈ R^n | g1 (x̄) ≥ 0, . . . , gs (x̄) ≥ 0} such that the quadratic modulegenerated by g1 , . . . , gs in R[X1 , . . . , Xn] is archimedean, and we provide a degree bound for the representation of a polynomial f ∈ R[X1 , . . . , Xn , Y] which is positive on S × R as an explicitelement of the quadratic module generated by g1 , . . . , gs in R[X1 , . . . , Xn , 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.