Polynomial bounds for Birch’s theorem

Let $K$ be a number field and let $P_1, … , P_s$ in $K[x_1, … , x_n]$ be forms of odd degrees. In 1957, Birch proved that if the number of variables n is sufficiently large then the forms have a non trivial zero in $K^n$. Apart from some small degrees, the bound on the number of variables required was of recursive type. We prove that, for any fixed degree, n may be taken polynomial in s. Joint work with Amichai Lampert and Andrew Snowden.