## Minimale Erzeugung von Positivbereichen.(English)Zbl 0546.14016

Let $$R$$ be a real closed field and $$V$$ an affine algebraic $$R$$-variety. A positive cone is a semialgebraic subset $$S\subset V(R)$$ of the form $$S=S(f_1,\ldots,f_s)=\{x\in V(R)\mid f_i(x)>0$$ for $$i=1,\ldots,s\}$$ with $$f_i\in R[V]$$. Let $$s(S)=\min\{s\mid S=S(f_1,\ldots,f_s)\}, s(V)=\sup\{s(S)\mid S$$ a positive cone in $$V\}$$ and $$s(n)=\sup\{s(V)\mid \dim V=n\}$$. It is shown that $$s(n)<\infty$$. In fact $$s(n)=n$$ for $$0<n\le 3$$. Also an explicit bound of $$s(n)$$ is given for general $$n$$ and a characterization of the positive cones among the semialgebraic sets.
Reviewer: Ludwig Bröcker

### MSC:

 14P10 Semialgebraic sets and related spaces 12D15 Fields related with sums of squares (formally real fields, Pythagorean fields, etc.) 11E10 Forms over real fields
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