Quadratic forms and systems of forms in many variables

TL;DR

This paper establishes an asymptotic formula for the number of integer solutions to systems of quadratic and higher-degree forms in many variables, advancing the understanding of the Hasse principle and solution counts.

Contribution

It proves an asymptotic formula for solutions to systems of forms in many variables, extending previous results and providing a framework for future applications of the circle method.

Findings

Asymptotic formula for quadratic forms in at least 9R variables

Extension to forms of degree d with bounds depending on d and R

Potential application of circle method to nonsingular systems

Abstract

Let $F_1,\dotsc,F_R$ be quadratic forms with integer coefficients in $n$ variables. When $n\geq 9R$ and the variety $V(F_1,\dotsc,F_R)$ is a smooth complete intersection, we prove an asymptotic formula for the number of integer points in an expanding box at which these forms simultaneously vanish, which in particular implies the Hasse principle for $V(F_1,\dotsc,F_R)$. Previous work in this direction required $n$ to grow at least quadratically with $R$. We give a similar result for $R$ forms of degree $d$, conditional on an upper bound for the number of solutions to an auxiliary inequality. In principle this result may apply as soon as $n> d2^dR$. In the case that $d\geq 3$, several strategies are available to prove the necessary upper bound for the auxiliary inequality. In a forthcoming paper we use these ideas to apply the circle method to nonsingular systems of forms with real coefficients.