Quartic Forms in Many Variables

TL;DR

This paper establishes new bounds on the number of variables needed for quartic, cubic, quadratic, and linear forms over p-adic fields to have non-trivial zeros, advancing understanding in algebraic number theory.

Contribution

It provides improved variable bounds for the existence of non-trivial zeros in quartic p-adic forms and systems of forms of lower degree, including specific results for cubic systems.

Findings

A quartic p-adic form with at least 3192 variables has a non-trivial zero.

Two cubic forms require at most 132 variables for a non-trivial zero.

New bounds are established for systems of quadratic and linear forms.

Abstract

We show that a quartic $p$-adic form with at least $3192$ variables possesses a non-trivial zero. We also prove new results on systems of cubic, quadratic and linear forms. As an example, we show that for a system comprising two cubic forms $132$ variables are sufficient.