Rational certificates of non-negativity on semialgebraic subsets of cylinders

Abstract

Let g1, . . . , gs ∈ R[X1, . . . , Xn, Y ] and S = {(¯x, y) ∈ R n+1 | g1(¯x, y) ≥ 0, . . . , gs(¯x, y) ≥ 0} be a non-empty, possibly unbounded, subset of a cylinder in R n+1. Let f ∈ R[X1, . . . , Xn, Y ] be a polynomial which is positive on S. We prove that, under certain additional assumptions, for any non-constant polynomial q ∈ R[Y ] which is positive on R, there is a certificate of the non-negativity of f on S given by a rational function having as numerator a polynomial in the quadratic module generated by g1, . . . , gs and as denominator a power of q.