Systems of forms in many variables

TL;DR

This paper establishes a new Hasse principle and asymptotic solution count for systems of homogeneous forms with many variables, improving previous bounds when the number of equations is at least two and degree at least four.

Contribution

It provides a nonsingular Hasse principle and asymptotic formula for solutions of systems of forms, requiring fewer variables than prior results for certain degrees and numbers of equations.

Findings

Proves a nonsingular Hasse principle for systems with n ≥ d2^d R + R variables.

Derives an asymptotic count of solutions in integers of bounded height.

Improves variable bounds for systems with R ≥ 2 and degree d ≥ 4.

Abstract

We consider systems $\vec{F}(\vec{x})$ of $R$ homogeneous forms of the same degree $d$ in $n$ variables with integral coefficients. If $n\geq d2^dR+R$ and the coefficients of $\vec{F}$ lie in an explicit Zariski open set, we give a nonsingular Hasse principle for the equation $\vec{F}(\vec{x})=\vec{0}$, together with an asymptotic formula for the number of solutions to in integers of bounded height. This improves on the number of variables needed in previous results for general systems $\vec{F}$ as soon as the number of equations $R$ is at least 2 and the degree $d$ is at least 4.