Geometric Embeddability of Complexes is $\exists \mathbb R$-complete

TL;DR

This paper proves that deciding whether a given abstract simplicial complex can be geometrically embedded in Euclidean space is computationally as hard as solving arbitrary polynomial equations over the reals, establishing its xists reals-completeness.

Contribution

It establishes the xists reals-completeness of the geometric embeddability problem for complexes in dimensions three and higher, providing the first hardness results for this problem.

Findings

The problem is xists reals-complete for all d  and k -1,d.

Deciding geometric embeddability is polynomial-time equivalent to solving polynomial equations over the reals.

The problem is NP-hard, indicating significant computational difficulty.

Abstract

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.