Polynomial Bounds for Birch's Theorem

TL;DR

The paper tackles the longstanding problem of bounding Birchs-type variable counts for systems of odd-degree forms by establishing polynomial-in-s bounds. It introduces a robust framework of relative strength and regularization to replace general forms with strong towers of polynomial size, then analyzes multi-linear Taylor expansions and polarization to transfer strength through subspace restrictions. A central technical core proves regularity results for towers via intricate rank controls (partition and geometric ranks) and gluing lemmas, enabling densification of rational points and the derivation of quantitative bounds. Together, these lead to polynomial bounds for N(d,s) and a Zariski-closure codimension bound that holds uniformly across Brauer fields and, in totally imaginary settings, for broader degrees. The approach harmonizes geometric regularity with number-field weak approximation to yield sharp, scalable results with potential reach beyond Birch-type problems.

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.

Theorems & Definitions (90)

• Theorem 1.1
• Definition 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Proposition 2.4
• proof