Representing Matroids over the Reals is ∃ R-complete

A matroid M is an ordered pair (E, Z), ) , where E is a finite set called the ground set and a collection Z C 2( E) called the independent sets which satisfy the conditions: (i) 0 is an element of Z, (ii) I ' C I E Z implies I ' is an element of Z, and (iii) I1, 1 , I-2 is an element of Z and I1 1 < I-2 implies that there is an e E I2 2 such that I1 1 U {e} is an element of Z. The rank rk (M) of a matroid M is the maximum size of an independent set. We say that a matroid M = (E, Z) ) is representable over the reals if there is a map phi : E R rk ( M ) such that I is an element of Z if and only if phi(I) forms a linearly independent set. We study the problem of M ATROID R-R EPRESENTABILITY 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 M ATROID R-R EPRESENTABILITY is XR-complete, already for matroids of rank 3 . The complexity class XR 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.