Small sunflowers and the structure of slice rank decompositions

TL;DR

This paper investigates the structure of tensor decompositions under slice rank, showing that certain sunflower configurations imply a more structured decomposition, with applications to bounding the number of such decompositions over finite fields.

Contribution

It establishes a structural result linking sunflower configurations in tensor decompositions to their canonical forms, and provides bounds on the number of decompositions over finite fields.

Findings

Sunflower structures in tensor decompositions imply a more canonical form.

Bounded the number of tensor decompositions with fixed slice rank over finite fields.

Provided a method to classify tensor decompositions based on sunflower configurations.

Abstract

Let $d \ge 3$ be an integer. We show that whenever an order-$d$ tensor admits $d+1$ decompositions according to Tao's slice rank, if the linear subspaces spanned by their one-variable functions constitute a sunflower for each choice of special coordinate, then the tensor admits a decomposition where these linear subspaces are contained in the centers of these respective sunflowers. As an application, we deduce that for every nonnegative integer $k$ and every finite field $\mathbb{F}$ there exists an integer $C(d,k,|\mathbb{F}|)$ such that every order-$d$ tensor with slice rank $k$ over $\mathbb{F}$ admits at most $C(d,k,|\mathbb{F}|)$ decompositions with length $k$, up to a class of transformations that can be easily described.