Two improvements in Birch's theorem on forms

TL;DR

This paper improves understanding of the size and density of solutions to systems of odd degree forms over Birch fields, showing the solution set is large and often dense under certain conditions.

Contribution

It demonstrates that the solution set's Zariski closure has bounded codimension and becomes dense when forms have high strength, extending recent work on forms over Birch fields.

Findings

Zariski closure of solutions has bounded codimension.

High strength forms lead to Zariski dense solution sets.

Results extend recent improvements on classical theorems.

Abstract

Let $K$ be a Birch field, that is, a field for which every diagonal form of odd degree in sufficiently many variables admits a non-zero solution; for example, $K$ could be the field of rational numbers. Let $f_1, \ldots, f_r$ be homogeneous forms of odd degree over $K$ in $n$ variables, and let $Z$ be the variety they cut out. Birch proved if $n$ is sufficiently large then $Z(K)$ contains a non-zero point. We prove two results which show that $Z(K)$ is actually quite large. First, the Zariski closure of $Z(K)$ has bounded codimension in $\mathbf{A}^n$. And second, if the $f_i$'s have sufficiently high strength then $Z(K)$ is in fact Zariski dense in $Z$. The proofs use recent results on strength, and our methods build on recent work of Bik, Draisma, and Snowden, which established similar improvements to Brauer's theorem on forms.