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

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
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