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.

翻译：拟阵$M$是一个有序对$(E,I)$，其中$E$是称为基集的有限集合，$I\subset 2^{E}$是称为独立集的集合，满足条件：(i) $\emptyset \in I$，(ii) $I'\subset I \in I$ 蕴含 $I'\in I$，且(iii) $I_1,I_2 \in I$ 且 $|I_1| < |I_2|$ 蕴含存在 $e\in I_2$ 使得 $I_1\cup \{e\} \in I$。拟阵$M$的秩$rank(M)$是独立集的最大大小。若存在映射$\varphi \colon E \rightarrow \mathbb{R}^{rank(M)}$使得$I\in I$当且仅当$\varphi(I)$构成线性无关集，则称拟阵$M=(E,I)$在实域上可表示。我们研究拟阵在实域上的可实现性问题。给定拟阵$M$，我们询问是否存在欧几里得空间中的一组点表示$M$。我们证明，即使对于秩为3的拟阵，拟阵可实现性也是$\exists \mathbb R$-完全的。复杂度类$\exists \mathbb R$可定义为与判定整数系数多元多项式是否存在实根多项式时间等价的算法问题族。我们的方法类似于文献中的先前方法，然而该结果本身从未被指出，且计算机科学文献中尚无现成证明。