Representing Matroids over the Reals is $\exists \mathbb R$-complete

Abstract

A matroid M is an ordered pair (E, I), where E is a finite set called the ground set and a collection I ⊂ 2 E called the independent sets which satisfy the conditions: (I1) ∅ ∈ I, (I2) I ⊂ I ∈ I implies I ∈ I, and (I3) I1, I2 ∈ I and |I1| < |I2| implies that there is an e ∈ I2 such that I1 ∪ {e} ∈ I. The rank rk(M ) of a matroid M is the maximum size of an independent set. We say that a matroid M = (E, I) is representable over the reals if there is a map ϕ : E → R rk(M ) such that I ∈ I if and only if ϕ(I) forms a linearly independent set.We study the problem of Matroid R-Representability over the reals. Given a matroid M , we ask whether there is a set of points in the Euclidean space representing M . We show that Matroid R-Representability is ∃R-complete, already for matroids of rank 3. The complexity class ∃R can be defined as the family of algorithmic problems that is polynomial-time equivalent to determining if a multivariate polynomial with integer coefficients has a real root.Our methods are similar to previous methods from the literature. Yet, the result itself was never pointed out and there is no proof readily available in the language of computer science.