Exact values of generic subrank

TL;DR

This paper determines the exact subrank of generic tensors in complex spaces, providing a precise formula and extending results to tensors of all orders and dimensions, thus resolving open questions in tensor theory.

Contribution

It establishes the exact subrank for generic tensors in $C^{n,n,n}$ and all tensor orders, answering previously open questions and computing related tensor variety dimensions.

Findings

Subrank of generic tensors in $C^{n,n,n}$ is $loor{ oot{3}n - 2}$.

General subrank formulas for tensors of all orders and dimensions.

Dimensions of tensor varieties with subrank at least $r$ are computed.

Abstract

In this article we prove the subrank of a generic tensor in $\mathbb{C}^{n,n,n}$ to be $Q(n) = \lfloor\sqrt{3n - 2}\rfloor$ by providing a lower bound to the known upper bound. More generally, we find the generic subrank of tensors of all orders and dimensions. This answers two open questions posed in arXiv:2205.15168v2. Finally, we compute dimensions of varieties of tensors of subrank at least $r$.