Strength is bounded linearly by Birch rank

TL;DR

This paper establishes a linear upper bound on the strength of homogeneous polynomials in terms of Birch rank across various fields, advancing understanding of polynomial complexity and its applications in number theory.

Contribution

It provides the first linear bound for forms of degree greater than three, linking strength and Birch rank, and applies this to improve results in the Hardy-Littlewood circle method and tensor analysis.

Findings

Strength is bounded linearly by Birch rank over multiple fields.

The result applies to forms of degree greater than three, partially resolving a conjecture.

Over finite fields, a quasi-linear bound for tensor partition rank in terms of analytic rank is achieved.

Abstract

Let $f$ be a homogeneous polynomial over a field. For many fields, including number fields and function fields, we prove that the strength of $f$ is bounded above by a constant multiple of the Birch rank of $f.$ The constant depends only on the degree of $f$ and the absolute transcendence degree of the field. This is the first linear bound obtained for forms of degree greater than three, partially resolving a conjecture of Adiprasito, Kazhdan and Ziegler. Our result has applications for the Hardy-Littlewood circle method. The circle method yields an asymptotic formula for counting integral zeros of (collections of) homogeneous polynomials, provided the Birch rank is sufficiently large -- a natural geometric condition. Our main theorem implies that these formulas hold even if we only assume a similar lower bound on the strength of the (collection of) homogeneous polynomials -- an arithmetic condition which is a priori weaker. This answers questions of Cook-Magyar and Skinner, and also yields a new proof of a seminal result of Schmidt as a consequence of Birch's earlier work. Over finite fields we obtain a quasi-linear bound for partition rank of tensors in terms of analytic rank, improving Moshkovitz-Zhu's state of the art bound.