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


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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