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On Bröcker’s \(t\)-invariant and separating families for constructible sets. (English) Zbl 0829.14026

The author continues his work in the theory of minimal generation of constructible sets in the real spectrum of a commutative ring. The main result gives upper bounds on the complexity of the constructible sets. Let \(X\) be a saturated and simply layered subset (see the definition in the paper) of the real spectrum of a commutative ring \(A\) of dimension \(d\), and let \(S \subseteq X\) be a constructible set described by a fixed separating family \(f_1, \ldots, f_n\). Then there exist invariants \(s_1, \ldots, s_d\) depending only on \(X\) and on elements \(f_1, \ldots, f_n\) such that
(1) \(S\) has a separating family with \(\leq \sum^d_{i = 1} p(s_i)\) elements, where \(p(s) = 4^s - 2^s + 1\).
(2) \(S\) is expressible as a union of \(\leq \sum^d_{i = 1} t(s_i)\) basis sets, where \(t(s) = s\) if \(t \leq 2\) and \(t(s)={4^{s - 1} - 2^{s-1} + 1\choose 2^{2s-3}-2^{s-2}+1}\), otherwise.
Another result states that if \(d(X) = 0\) then \((X,A^*/T^*)\) is a space of orderings.
Reviewer: M.Kula (Katowice)

MSC:

14P10 Semialgebraic sets and related spaces
68Q25 Analysis of algorithms and problem complexity
12D15 Fields related with sums of squares (formally real fields, Pythagorean fields, etc.)
11E10 Forms over real fields
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References:

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