Stability index of real varieties
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). \textit{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.\textit{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.
scientific article; zbMATH DE number 4177288