Geometry of multilinear varieties over infinite fields and its applications

TL;DR

This paper develops a geometric framework for multilinear varieties over infinite fields, proving fundamental results that resolve several conjectures related to tensor and polynomial ranks.

Contribution

It establishes a codimension formula and existence of high-dimensional irreducible subvarieties, providing a foundation for analyzing various tensor and polynomial ranks over infinite fields.

Findings

Resolved the Adiprasito-Kazhdan-Ziegler conjecture on partition rank stability.

Settled the conjecture on the equivalence of strength and Birch rank over infinite fields.

Strengthened existing theorems on multilinear varieties over infinite fields.

Abstract

Multilinear varieties, defined as the sets of rational points of varieties cut out by multilinear functions, were first introduced and studied by Gowers and Mili\'{c}evi\'{c}[Proc. Edinb. Math. Soc., 2021] for finite $\mathbb{K}$. In this paper, we investigate multilinear varieties over infinite fields from a geometric perspective. We establish two fundamental results: a codimension formula for the Zariski closure of a multilinear variety, and the existence of a high-dimensional irreducible subvariety passing through any given $\mathbb{K}$-rational point. These results serve as a geometric foundation for analyzing various ranks of tensors and homogeneous polynomials, including partition rank, analytic rank, geometric rank, (collective) strength and (collective) Birch rank. As applications, we resolve the Adiprasito-Kazhdan-Ziegler conjecture [arXiv:2102.03659, 2021] on the stability of partition rank for perfect infinite fields. We thereby settle the stability conjecture for collective strength [Selecta Math., 2024], as well as the conjecture on the linear equivalence between strength and Birch rank [arXiv:2410.00248, 2024] for such fields. Moreover, our results immediately yield a strengthening of the theorems of Bik-Draisma-Snowden [arXiv:2401.02067, 2024] and Lampert-Snowden [arXiv:2406.18498, 2024], for multilinear varieties over infinite fields.