Zeros of p-adic forms

TL;DR

This paper develops a variant of Brauer's induction method to establish the existence of non-trivial zeros in p-adic forms, showing that quartic forms with sufficiently many variables always have zeros, with improvements for odd primes.

Contribution

Introduces a new variant of Brauer's induction method and proves that quartic p-adic forms with at least 9127 variables have non-trivial zeros, with improved bounds for odd p.

Findings

Quartic p-adic forms with ≥9127 variables have non-trivial zeros.

Fewer variables are needed for odd p.

Includes new results on quintic forms and systems of forms.

Abstract

A variant of Brauer's induction method is developed. It is shown that quartic p-adic forms with at least 9127 variables have non-trivial zeros, for every p. For odd p considerably fewer variables are needed. There are also subsidiary new results concerning quintic forms, and systems of forms.