Descent problem for certificate of non-negativity on semi-algebraic sets

Abstract

Let $F$ be a subfield of $\mathbb R$ and let $K$ be a basic closed semi-algebraic set in $\mathbb R$ with $\partial K\subset F$. Let $\mathcal N$ be the natural choice of generators of $K$. We show that if $f\in F[x]$ is $\geq 0$ on $K$, then $f$ can be written as $$f=\sum_{e\in\{0,1\}^s } a_e\sigma_e g^e $$ where $a_e\in F_{\geq 0}$, $\sigma_e\in \sum F[x]^2$ and $g^{e}=g_1^{e_1} \cdots g_s^{e_s}$. In other words, the preordering $T_{\mathcal N}$ of $F[x]$ is saturated. In case $F=\mathbb R$, this result is due to Kuhlmann and Marshall. As an application, we prove that if $K$ is compact, then $M_{\mathcal N}=T_{\mathcal N}=Pos(K)$. In other words, the quadratic module $M_{\mathcal N}$ of $F[x]$ is saturated.