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Stability index of real varieties. (English) Zbl 0715.14049

Let R be a real closed field, V an affine (non-necessarily irreducible) R-variety and \(f_ 1,...,f_ r\in \Gamma (V,{\mathcal O}_ V)\) polynomial functions on V. If \(S(f_ 1,...,f_ r)\) is the semi-algebraic set of points of V in which \(f_ 1,...,f_ r\) are strictly positive and \(\bar S(f_ 1,...,f_ r)\) is the set of points in which \(f_ 1,...,f_ r\) are non-negative then the sets \(S(f_ 1,...,f_ r)=S\) are called basic open sets and the sets of the form \(F=\bar S(f_ 1,...,f_ r)\) are called basic closed sets on V. Let s(S) be (resp. \(s(F)\) the minimal number of inequalities necessary to describe S (resp. F). Then the supremum s(V) of all the numbers s(S) when S are non empty basic open sets on V is finite and it is called the geometric stability index of the variety V. Also, \(\bar s(S)\) is defined as the supremum of s(F). L. Bröcker proved that there are upper bounds for the numbers s(V), \(\bar s(V)\), which depend on the dimension of V. More precisely, in a preprint L. Bröcker showed that if \(\dim(V)=n>0\) then \(n+2\leq \bar s(V)\leq n(n+1)\) for \(n\geq 3\) and \(\bar s(V)=n(n+1)\) for \(n=1\) or \(n=2.\)
In the present paper this result is reproved but it is also presented the following complete result: if \(n>0\) then \(\bar s(V)=n(n+1)\). - The method to prove this is to obtain by an inductive argument that one has also \(\bar s(V)\geq n(n+1).\)
On the geometric stability index s(V), L. Bröcker showed that \(s(V)=\dim (V)=n\) if \(1\leq n\leq 3\) and that in general \(n\leq s(V)\leq 3\cdot 2^{m-1}\) if \(n=2m\), \(n\leq s(V)\leq 2^ m\) if \(n=2m-1\). In the present paper it is established the following precise result: if V is a real n-dimensional variety, \(n>0\), then \(s(V)=n\) (theorem 2, corollary 4). To prove this result, the author utilises: 1. the real spectrum of a ring developed by M. Coste and M.-F. Roy in 1982 and other authors in more recent works. 2.M. Marshall’s theory of spaces of orderings [Trans. Am. Math. Soc. 258, 505-521 (1980; Zbl 0427.10015)] in which theory the author had obtained new useful results.
Reviewer: M.I.Becheanu

MSC:

14P10 Semialgebraic sets and related spaces
12D15 Fields related with sums of squares (formally real fields, Pythagorean fields, etc.)
26C05 Real polynomials: analytic properties, etc.

Citations:

Zbl 0427.10015
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References:

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