Polynomial Bounds for Birch's Theorem

Amichai Lampert; Andrew Snowden; Tamar Ziegler

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arXiv · 2512.00697

Polynomial Bounds for Birch's Theorem

Amichai Lampert·Andrew Snowden·Tamar Ziegler

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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.

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Amichai Lampert, Andrew Snowden, Tamar Ziegler. 2025-11-30. Polynomial Bounds for Birch's Theorem. https://arxiv.org/abs/2512.00697

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