A note on p-adic solubility for forms in many variables

TL;DR

This paper introduces a new approach to analyze p-adic solutions for forms in many variables, showing that the existence of solutions ensures positive singular series, leading to asymptotic formulas for zeros of cubic forms.

Contribution

It presents a novel method that simplifies establishing p-adic solution positivity, enabling asymptotic analysis for cubic forms in at least 131 variables.

Findings

Established positivity of the singular series from p-adic solutions

Derived an asymptotic formula for zeros of cubic forms in many variables

Developed a version of Hensel's Lemma for linear spaces

Abstract

By adopting a new approach to the analysis of the density of p-adic solutions arising in applications of the circle method, we show that under modest conditions the existence of non-trivial p-adic solutions suffices to establish positivity of the singular series. This improves on earlier approaches due to Davenport, Schmidt and others, and allows us to establish an asymptotic formula for the number of simultaneous zeros of non-singular pairs of cubic forms in at least 131 variables. As a by-product, we obtain a version of Hensel's Lemma for linear spaces.