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According to the Universality Theorem by Mnev (see below theorem 8.6.6 from [1]), for any open semialgebraic variety V there is a uniform oriented matroid of rank 3 whose realization space is stably equivalent to V.

My question is whether one can transfer the result also to uniform oriented matroids of rank 4 and higher, i.e., for a fixed r $\ge$ 4 and for any open semialgebraic variety V is there a uniform oriented matroid of rank r whose realization space is stably equivalent to V?

8.6.6 Universality Theorem (Mnev 1988).

  1. Let $V \subset \mathbb{R}^s$ be any semialgebraic variety. Then there exists a rank 3 oriented matroid M. whose realization space R(M) is stably equivalent to V.
  2. If V is an open subset of $\mathbb{R}^s$, then M may be chosen to be uniform.

[1] A. Björner, M. L. Vergnas, B. Sturmfels, N. White, and G. M. Ziegler, Oriented Matroids. Cambridge University Press, 1999.

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Yes, the Universality Theorem works for any rank $r\geq3$. One way to prove this is to consider a rank $3$ oriented matroid $M$ with the desired realization space, then take the dual $M^*$, do a lexicographic extension in general position and then take the dual again. Repeat this $r-3$ times.

Lexicographic extensions preserve the realization space up to stable equivalence, and so does taking the dual. A lexicographic extension increases the number of elements by one and keeps the rank. Hence, doing it to the dual increases the rank of the primal by one.

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  • $\begingroup$ Thanks for the answer, Arnau! I have two follow-up questions. Will the procedure you suggested run in polynomial time? Is the procedure a known technique for which there is a literature reference? Thanks! $\endgroup$
    – Jae
    Commented Nov 17, 2014 at 14:50
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    $\begingroup$ Yes, this should run in polynomial time. Lexicographic extensions are explained in section 7.2 of [1]. In Lemma 8.2.1 and Proposition 8.2.2 in the same book it is proven that the realization space of a lexicographic extension is preserved up to homotopy equivalence. The proof for stable equivalence is similar. You can also have a look at arxiv.org/abs/1402.7207. $\endgroup$
    – Arnau
    Commented Nov 18, 2014 at 11:51
  • $\begingroup$ Really cool and simple trick, thx @Arnau ! I had something more complicate in mind but i think equivalent. $\endgroup$
    – Nikolai Mnev
    Commented Mar 31, 2019 at 14:27

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