Polynomial Bounds for Birch's Theorem

Let $K$ be a number field and $f_1,\dots ,f_s\in K[x_1,\dots ,x_n]$ 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.