Two improvements in Birch's theorem on forms

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.