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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
##### Keywords:
real holomorphy ring; preordering
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