Representations of integers by systems of three quadratic forms

TL;DR

This paper advances the understanding of representing integers with three quadratic forms by reducing the variable count needed for asymptotic formulas, employing a novel three-dimensional circle method under nonsingularity conditions.

Contribution

It introduces a three-dimensional analogue of Kloosterman's circle method, lowering the variable threshold to k ≥ 10 for almost all tuples of integers.

Findings

Achieved asymptotic formulas for representations with fewer variables.

Developed a new geometric approach for systems of three quadratic forms.

Extended classical circle method techniques to three dimensions.

Abstract

It is classically known that the circle method produces an asymptotic for the number of representations of a tuple of integers $(n_1,\ldots,n_R)$ by a system of quadratic forms $Q_1,\ldots, Q_R$ in $k$ variables, as long as $k$ is sufficiently large; reducing the required number of variables remains a significant open problem. In this work, we consider the case of 3 forms and improve on the classical result by reducing the number of required variables to $k \geq 10$ for "almost all" tuples, under appropriate nonsingularity assumptions on the forms $Q_1,Q_2,Q_3$. To accomplish this, we develop a three-dimensional analogue of Kloosterman's circle method, in particular capitalizing on geometric properties of appropriate systems of three quadratic forms.