Mnëv’s universality theorem revisited. (English) Zbl 0855.05041

Richter-Gebert, Jürgen

Sémin. Lothar. Comb. 34, B34h, 15 p. (1995).

Summary: This article presents a complete proof of Mnëv’s universality theorem and a complete proof of Mnëv’s universal partition theorem for oriented matroids. The universality theorem states that, for every primary semialgebraic set \(V\) there is an oriented matroid \(M\), whose realization space is stably equivalent to \(V\). The universal partition theorem states that, for every partition \(V\) of \(\mathbb{R}^n\) indiced by \(m\) polynomial functions \(f(1),\dots, f(n)\) with integer coefficients, there is a corresponding family of oriented matroids \((M(s))\), with \(s\) ranging in the set of \(m\)-tuples with elements in \(\{- 1, 0, + 1\}\), such that the collection of their realization spaces is stably equivalent to the family \(V\).

MSC:

05B35

Combinatorial aspects of matroids and geometric lattices

Keywords:

universality theorem; universal partition theorem; matroids; polynomial; realization spaces