Zeros of p-adic forms

TL;DR

This paper proves that all sufficiently high-variable p-adic quintic forms have non-trivial zeros and presents new results on systems of quadratic and cubic forms.

Contribution

It establishes a new bound for the number of variables ensuring non-trivial zeros of p-adic quintic forms and advances understanding of systems involving quadratic and cubic forms.

Findings

All p-adic quintic forms with more than 4,562,911 variables have a non-trivial zero.

New results on the solvability of systems of quadratic and cubic forms.

Improved bounds and conditions for the existence of solutions in p-adic forms.

Abstract

We show that all $p$-adic quintic forms in at least $n>4562911$ variables have a non-trivial zero. We also derive new result concerning systems of cubic and quadratic forms.