On Bröcker's \(t\)-invariant and separating families for constructible sets
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.
minimal generation of constructible sets
scientific article; zbMATH DE number 699784