$p$-adic Zeros of Quintic Forms

TL;DR

This paper proves that quintic forms over p-adic fields with sufficiently many variables have non-trivial zeros, given the residue field size exceeds a certain threshold, advancing understanding of solutions to polynomial equations in number theory.

Contribution

It establishes a new bound on the number of variables needed for quintic forms over p-adic fields to have non-trivial zeros, under specific residue field conditions.

Findings

Quintic forms with at least 26 variables over p-adic fields have non-trivial zeros.

The result holds when the residue field size exceeds 9.

This extends previous bounds for polynomial forms over local fields.

Abstract

It is shown that a quintic form over a p-adic field with at least 26 variables has a non-trivial zero, providing that the cardinality of the residue class field exceeds 9.