Representing Matroids over the Reals is [Formula presented]-complete

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: (i) ∅ ∈ I, (ii) I^′ ⊂ I ∈ I implies I^′ ∈ I, and (iii) I₁, I₂ ∈ I and |I₁| < |I₂| implies that there is an e ∈ I₂ such that [Formula presented]. 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 [Formula presented] such that [Formula presented] if and only if φ(I) forms a linearly independent set. We study the problem of Matroid [Formula presented]-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 [Formula presented]-Representability is [Formula presented]-complete, already for matroids of rank 3. The complexity class [Formula presented] 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.