## 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.

Zbl 0427.10015
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### References:

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