Submission date: 8 November 2018

Abstract:

We prove that, under some additional assumption, Putinar's Positivstellensatz holds on cylinders of type S × ℝ with S = { x in ℝ^n | g_1(x) ≥ 0, ..., g_s(x) ≥ 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 ∈ ℝ[X_1, ..., X_n, Y] which is positive on S × 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.

Mathematics Subject Classification: 12D15, 13J30, 14P10

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Full text arXiv 1811.03586: pdf, ps.