Séminaire Lotharingien de Combinatoire, B34h (1995), 15pp.

Jürgen Richter-Gebert

Mnëv's Universality Theorem Revisited

Abstract. 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.

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