Closure of the sums of 2d-powers in topological commutative rings

Details Closure of the sums of 2d-powers in topological commutative rings Closure of the sums of 2d-powers in topological commutative rings

2011-10-13

arxiv.org/abs/1110.3016v2

Mathematics

Rings and Algebras

Scientific paper

Let $R$ be a unital algebra over reals and $K\subseteq Hom(R,\mathbb{R})$, closed with respect to the product topology. We consider $R$ endowed with the topology induced by the family of seminorms $\rho_{\alpha}(a):=|\alpha(a)|$, for $\alpha\in K$ and $a\in R$. In case $K$ is compact, we also consider the topology induced by $\|a\|_K:=\sup_{\alpha\in K}|\alpha(a)|$ for $a\in R$. If $K$ is Zariski dense, then those topologies are Hausdorff. In this paper we prove that the closure of the cone of sums of 2d-powers, with respect to those two topologies is equal to the cone $\{a\in R:\alpha(a)\ge0,\forall\alpha\in K\}$. In particular, any continuous linear functional $L$ on the polynomial ring $R=\mathbb{R}[X_1,...,X_n]$ with $L(h^{2d})\ge0$ for each polynomial $h$, is integration with respect to a positive Borel measure supported on $K$. We give necessary and sufficient conditions to ensure the continuity of a linear functional. We compare our results with results of Schm\"udgen, Berg et al. and Lasserre.

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