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 $E$ 是一个称为基础集的有限集，而 $I\subset 2^{E}$ 是一个称为独立集的集合族，其满足以下条件：(i) $\emptyset \in I$，(ii) $I'\subset I \in I$ 蕴含 $I'\in I$，以及 (iii) $I_1,I_2 \in I$ 且 $|I_1| < |I_2|$ 蕴含存在一个 $e\in I_2$ 使得 $I_1\cup \{e\} \in I$。拟阵 $M$ 的秩 $rank(M)$ 是一个独立集的最大大小。我们称一个拟阵 $M=(E,I)$ 在实数域上可表示，如果存在一个映射 $\varphi \colon E \rightarrow \mathbb{R}^{rank(M)}$，使得 $I\in I$ 当且仅当 $\varphi(I)$ 构成一个线性无关集。我们研究拟阵在实数域上的可实现性问题。给定一个拟阵 $M$，我们询问是否存在欧几里得空间中的一个点集来表示 $M$。我们证明，拟阵可实现性问题是 $\exists \mathbb R$-完全的，即使对于秩为 3 的拟阵也是如此。复杂度类 $\exists \mathbb R$ 可以定义为这样一类算法问题族：它们在多项式时间内等价于判定一个具有整数系数的多元多项式是否存在实根。我们的方法与文献中先前的方法类似。然而，该结果本身从未被明确指出，并且在计算机科学的语言中尚无现成的证明。