Slice rank and analytic rank for trilinear forms

TL;DR

This paper provides an elementary proof that the slice rank of a trilinear form over a finite field is linearly bounded by its analytic rank, avoiding complex geometric invariant theory methods.

Contribution

It introduces a new, simpler proof connecting slice rank and analytic rank for trilinear forms, using coordinate fixing techniques.

Findings

Slice rank is linearly bounded by analytic rank.

Elementary proof avoids geometric invariant theory.

Coordinates fixing yields the decomposition.

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.