Countably many asymptotic tensor ranks

TL;DR

This paper proves that the set of all possible asymptotic subranks of complex tensors, which are algebraic invariants, is countable, settling a question related to tensor rank gaps.

Contribution

It establishes that all algebraic tensor invariants' asymptotic subranks form a countable set, confirming a conjecture in tensor theory.

Findings

The set of asymptotic subranks for algebraic tensor invariants is countable.

The result applies to all invariants invariant under complex field automorphisms.

This settles an open question about the structure of tensor invariants.

Abstract

In connection with recent work on gaps in the asymptotic subranks of complex tensors the question arose whether the number of nonnegative real numbers that arise as the asymptotic subrank of some complex tensor is countable. In this short note we settle this question in the affirmative, for all tensor invariants that are algebraic in the sense that they are invariant under field automorphisms of the complex numbers.