Mnëv’s Universality Theorem revisited

This article presents a complete proof of Mnëv’s Universality Theorem and and a first 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 IR induced by m polynomial functions f1, . . . , fm with integer coefficients there is a corresponding family of oriented matroids (Mσ)σ∈{−1,0,+1}m such that the collection of their realization spaces is stably equivalent to the family V.