Mnëv's Universality Theorem Revisited

Jürgen Richter-Gebert

Séminaire lotharingien de combinatoire, Tome 34 (1995)

Résumé

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 Rⁿ indiced by m polynomial functions f(1),...,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.