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?**




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**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.



The Universality Theorem by Mnev for uniform oriented matroids of rank 4 and higher
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.


