A version of Putinar's Positivstellensatz for cylinders

Abstract

We prove that, under some additional assumption, Putinar's Positivstellensatz holds on cylinders of type $S\times \mathbb{R}$ with $S=\{x\in {\mathbb{R}}^n|g_1(x)\ge 0,...,g_s(x)\ge 0\}$ such that the quadratic module generated by $g_1,...,g_s$ in $\mathbb{R}[X_1,...,X_n]$ is archimedean, and we provide a degree bound for the representation of a polynomial $f\in \mathbb{R}[X_1,...,X_n,Y]$ which is positive on $S\times \mathbb{R}$ as an explicit element of the quadratic module generated by $g_1,...,g_s$ in $\mathbb{R}[X_1,...,X_n,Y]$. We also include an example to show that an additional assumption is necessary for Putinar's Positivstellensatz to hold on cylinders of this type.

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