Polynomial Bounds for Birch's Theorem

Amichai Lampert, Andrew Snowden, Tamar Ziegler

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 nontr…