zbMATH — the first resource for mathematics

A real holomorphy ring without the Schmüdgen property. (English) Zbl 0971.12002
Let \(B_T(A)\) denote the real holomorphy ring determined by a preordering \(T\) of a real algebra \(A\). From papers of K. Schmüdgen [Math. Ann. 289, 203-206 (1991; Zbl 0744.44008)] and T. Wörmann [Strikt positive Polynome in der algebraischen Geometrie, PhD Thesis, Dortmund (1998; Zbl 1042.14037)] it follows that if \(B_T(A)=A\) then \(f>0\) on \(\text{Sper}_T(A)\) implies \(f\in T\). In the paper under review a preordering \(T\) in \(A=\mathbb{R}[t_1,t_2, \dots]\) (countable many variables) is constructed such that \(B_T(A)=A\) and if \(f>0\) on \(\text{Sper}_TA\) then \(f\in \mathbb{R}\). A similar property has the preordering \(\sum C^2\) (the set of sums of squares) in some multiquadratic extensions \(C\) of \(A\).

12D15 Fields related with sums of squares (formally real fields, Pythagorean fields, etc.)
14P10 Semialgebraic sets and related spaces
44A60 Moment problems
Full Text: DOI