The Complexity of Tensor Rank

TL;DR

This paper establishes that determining tensor rank over a field is as computationally complex as solving the existential theory of that field, linking tensor problems to fundamental algebraic decision problems.

Contribution

It proves the computational hardness of tensor rank determination over any field, extending previous NP-hardness results and revealing algebraic universality.

Findings

Tensor rank decision problem is as hard as the existential theory of the field.

The hardness proof generalizes NP-hardness to all fields.

The results connect tensor problems with fundamental algebraic decision problems.

Abstract

We show that determining the rank of a tensor over a field has the same complexity as deciding the existential theory of that field. This implies earlier NP-hardness results by H{\aa}stad~\cite{H90}. The hardness proof also implies an algebraic universality result.