Slice rank and analytic rank for trilinear forms

TL;DR

The paper addresses the relation between slice rank and analytic rank for trilinear forms over finite fields. It provides an elementary, GIT-free proof of the bound $srk(f) <= 5 ark(f) + 4 log_q(ark(f)+1) + 29$, and shows that the linear forms in the slice-rank decomposition can be obtained by fixing coordinates. The approach uses a probabilistic subspace argument to locate a large subspace where the fixed-coordinate bilinear forms have controlled rank, together with a Shpilka-Haramaty lemma and an induction on a rank parameter to conclude the bound. This work connects analytic rank to partition rank notions and highlights asymmetries in the current method, while discussing potential symmetric bounds and broader conjectures in the area.

Abstract

In this note, we present an elementary proof of the fact that the slice rank of a trilinear form over a finite field is bounded above by a linear expression in the analytic rank. The existing proofs by Adiprasito-Kazhdan-Ziegler and Cohen-Moshkovitz both rely on results of Derksen via geometric invariant theory. A novel feature of our proof is that the linear forms appearing in the slice rank decomposition are obtained from the trilinear form by fixing coordinates.

Theorems & Definitions (11)

• Definition 1.1
• Theorem 1.2
• Definition 1.3
• Conjecture 1.4
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof