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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
*R*^{n} 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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