A Weak Chevalley-Warning Theorem for Quasi-finite Fields

TL;DR

The paper proves that over quasi-finite fields, high-dimensional projective hypersurfaces of fixed degree always have rational points, extending classical results to a broader class of fields.

Contribution

It establishes a weak Chevalley-Warning type theorem for quasi-finite fields, showing existence of rational points on large enough hypersurfaces.

Findings

Existence of rational points on high-dimensional hypersurfaces over quasi-finite fields

A function f(d) bounds the dimension for guaranteed rational points

Extension of classical theorems to fields with finitely generated Galois groups

Abstract

There exists a function f: N -> N such that for every positive integer d, every quasi-finite field K and every projective hypersurface X of degree d and dimension at least f(d), the set X(K) is non-empty. This is a special case of a more general result about intersections of hypersurfaces of fixed degree in projective spaces of sufficiently high dimension over fields with finitely generated Galois groups.