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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 \(A\).

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