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

Mikkel Abrahamsen; Linda Kleist; Tillmann Miltzow

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arXiv · 2108.02585

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

Mikkel Abrahamsen·Linda Kleist·Tillmann Miltzow

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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.

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BibTeX RIS

Mikkel Abrahamsen, Linda Kleist, Tillmann Miltzow. 2021-08-05. Geometric Embeddability of Complexes is $\exists \mathbb R$-complete. https://arxiv.org/abs/2108.02585

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