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

TL;DR

This work proves that Matroid $\mathbb{R}$-Representability is $\exists \mathbb{R}$-complete, already for rank-$3$ matroids, connecting matroid realizability to the existential theory of the reals. It provides two distinct proofs: a self-contained reduction from Distinct-ETR using von Staudt-style arithmetic gadgets encoded in matroid independence constraints, and a route via order-type realizability by simulating order types with matroids. Central to the constructions are free helper points that avoid accidental incidences, and gadgets that implement addition, multiplication, and strict inequalities purely through matroid data. The results highlight a deep computational distinction between real and finite-field representability and illuminate the complexity of geometric realizability problems. Together, these findings bridge matroid theory, computational geometry, and real algebraic geometry, informing both theory and algorithmic approaches to matroid realization problems.

Abstract

A matroid $M$ is an ordered pair $(E,I)$, where $E$ is a finite set called the ground set and a collection $I\subset 2^{E}$ called the independent sets which 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 that there is an $e\in I_2$ such that $I_1\cup \{e\} \in I$. The rank $rank(M)$ 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 $\varphi \colon E \rightarrow \mathbb{R}^{rank(M)}$ 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 space representing $M$. We show that matroid realizability is $\exists \mathbb R$-complete, already for matroids of rank 3. The complexity class $\exists \mathbb R$ can be defined as the family of algorithmic problems that is polynomial-time is equivalent to determining if a multivariate polynomial with integers 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.

Figures (6)

• Figure 1: As $\ell$ separates $f$ from the other points, a projective transformation sending the line $\ell$ to infinity will flip the orientation of the triangles involving $f$ while keeping the other orientations unchanged.
• Figure 2: Encoding addition geometrically.
• Figure 3: Encoding addition geometrically.
• Figure 4: Encoding multiplication geometrically.
• Figure 5: Forcing $c$ to be on the same side of $\ell(a,b)$ as $e$. Rays ending at a point denote the strict inequality constraints.

Theorems & Definitions (16)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 4
• proof
• Lemma 5: folklore
• Lemma 6
• Lemma 7
• Lemma 8
• Lemma 9