We show that the decision problem of determining whether a given (abstract simplicial) $k$-complex has a geometric embedding in $\mathbb R^d$ is complete for the Existential Theory of the Reals for all $d\geq 3$ and $k\in\{d-1,d\}$. This implies that the problem is polynomial time equivalent to determining whether a polynomial equation system has a real solution. Moreover, this implies NP-hardness and constitutes the first hardness results for the algorithmic problem of geometric embedding (abstract simplicial) complexes.

翻译：我们显示,确定给定(抽象的)$k$-complex是否在$\mathbb R ⁇ d$中嵌入几何分数的决定问题,对于所有$d\geq 3美元和$k\in ⁇ d-1,d ⁇ $来说,对于真实存在的3美元和$k\in ⁇ d-1,d ⁇ $来说,已经完全解决了。这意味着问题是一个多元时,相当于确定一个多元方程系统是否有真正的解决办法。此外,这意味着NP-硬性,是几何嵌入(抽象的)复合体的算法问题的第一个硬性结果。