Weyl's inequality and systems of forms

TL;DR

This paper extends Weyl's inequality to general systems of forms, enabling the derivation of Hardy-Littlewood asymptotics for integer solutions of quadratic and cubic systems under weaker conditions, and discusses higher degree forms.

Contribution

Introduces a variant of Weyl's inequality applicable to general systems of forms, relaxing previous rank and non-singularity conditions for asymptotic analysis.

Findings

Established Hardy-Littlewood asymptotics for quadratic and cubic forms.

Weaker rank conditions than previous results.

Discussed implications for higher degree forms.

Abstract

By providing a variant of Weyl's inequality for general systems of forms we establish the Hardy-Littlewood asymptotic formula for the density of integer zeros of systems of quadratic or cubics forms under weaker rank conditions than previously known. We also briefly discuss what happens for systems of higher degree forms, and slightly relax the non-singularity condition in Birch's paper on forms in many variables.