Polynomial Bounds for Birch's Theorem

TL;DR

This paper improves bounds on the number of variables needed for Birch's theorem, showing it can be polynomial in the number of forms for fixed degrees, and extends results to totally imaginary fields.

Contribution

It establishes polynomial bounds for the number of variables in Birch's theorem and extends the results to forms of any degree over totally imaginary fields.

Findings

Number of variables can be polynomial in the number of forms for fixed degrees.

Zariski closure of rational zeros has polynomially bounded codimension.

Results hold for forms of any degree over totally imaginary fields.

Abstract

Let $K$ be a number field and $f_1,\ldots,f_s\in K[x_1,\ldots,x_n]$ forms of odd degrees. In 1957, Birch proved that if $n$ is sufficiently large then the forms always have a nontrivial zero in $K^n$. Apart from some small degrees, the number of variables required was so large that it has been described as "not even astronomical". We prove that, for any fixed degree, $n$ may be taken polynomial in $s$. We deduce this from a stronger result -- the Zariski closure of the set of rational zeros has codimension bounded by a polynomial in $s$. When $K$ is totally imaginary, our results hold for forms of any (possibly even) degrees.