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: (I1) $\emptyset \in I$, (I2) $I'\subset I \in I$ implies $I'\in I$, and (I3) $I_1,I_2 \in I$ 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 $\varphi : E \rightarrow \mathbb{R}^{rank(M)}$ 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：一个拟阵$M$是一个有序对$(E,I)$，其中$E$是称为基集的有限集，$I\subset 2^{E}$是称为独立集的子集族，满足以下条件：(I1) $\emptyset \in I$，(I2) 若$I'\subset I \in I$则$I'\in I$，(I3) 若$I_1,I_2 \in I$且$|I_1| < |I_2|$，则存在$e\in I_2$使得$I_1\cup \{e\} \in I$。拟阵$M$的秩$rank(M)$是独立集的最大大小。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$\varphi : E \rightarrow \mathbb{R}^{rank(M)}$，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组点来表示$M$。我们证明，即使对于秩为3的拟阵，拟阵的可实现性也是$\exists \mathbb R$-完全的。复杂度类$\exists \mathbb R$可以定义为与判定一个整数系数的多元多项式是否存在实根在多项式时间上等价的算法问题族。我们的方法类似于文献中的先前方法。然而，这一结果本身从未被指出过，并且目前没有现成的以计算机科学语言表述的证明可用。