Exact values of generic subrank

TL;DR

The paper determines exact generic subrank values for tensors of all orders and shapes over infinite fields, resolving open questions about when and how high a subrank is generically achievable. It reduces the problem to the generic rank of a structured projection matrix $M$ and develops a combinatorial crossing-out algorithm to guarantee a nonzero minor, thereby proving tight lower bounds that match prior upper bounds. Consequently, it establishes the exact formulas $Q(n_1,\ldots,n_k)=\min\{n_1,\ldots,n_k,\left\lfloor (n_1+\cdots+n_k-(k-1))^{1/(k-1)}\right\rfloor\}$ and, in the cubic case, $Q(n)=\left\lfloor \sqrt{3n-2} \right\rfloor$, with corresponding dimension results for subrank-at-least-$r$ varieties. The results extend to higher-order tensors and yield explicit dimension expressions $\dim(\mathcal{C}_r^{(n_1,\ldots,n_k)})=\prod n_i - r\bigl(r^{k-1}-\sum n_i+(k-1)\bigr)$ in characteristic $0$, opening avenues for generic-rank and border-subrank questions through algebraic-geometry and combinatorics.

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 $r$.

Figures (1)

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Theorems & Definitions (41)

• Corollary 1.0
• Theorem 1.1
• Theorem 1.1
• Theorem 1.1
• Remark 2.1
• Theorem 2.2: Theorem 3.9 10.4230/LIPIcs.CCC.2022.9
• Definition 2.3
• Lemma 2.4
• proof
• Example 2.5