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 calledthe ground set and a collection $I\subset 2^{E}$ called the independent setswhich satisfy the conditions: (i) $\emptyset \in I$, (ii) $I'\subset I \in I$implies $I'\in I$, and (iii) $I_1,I_2 \in I$ and $|I_1| < |I_2|$ implies thatthere is an $e\in I_2$ such that $I_1\cup \{e\} \in I$. The rank $rank(M)$ of amatroid $M$ 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 $\varphi \colon E\rightarrow \mathbb{R}^{rank(M)}$ such that $I\in I$ if and only if$\varphi(I)$ forms a linearly independent set. We study the problem of matroid realizability over the reals. Given a matroid$M$, we ask whether there is a set of points in the Euclidean spacerepresenting $M$. We show that matroid realizability is \exists \mathbbR-complete, already for matroids of rank 3. The complexity class $\exists\mathbb R$ can be defined as the family of algorithmic problems that ispolynomial-time is equivalent to determining if a multivariate polynomial withintegers coefficients has a real root. Our methods are similar to previous methods from the literature. Yet, theresult itself was never pointed out and there is no proof readily available inthe language of computer science.Comment: v2 and v3: Minor changes v4: Final version, to appear in DMTC