A two-dimensional delta symbol method and its application to pairs of quadratic forms

TL;DR

This paper introduces a two-dimensional delta symbol method to improve the circle method for counting integral points on quadratic forms, reducing variables needed under certain hypotheses.

Contribution

The paper develops a novel two-dimensional delta symbol technique and applies it to quadratic forms, achieving asymptotic formulas with fewer variables than previous methods.

Findings

Established asymptotic formula for integral points on quadratic forms with at least 10 variables.

Under the Generalized Lindelöf Hypothesis, reduced variables to 9 using double Kloosterman refinement.

Heuristic suggests the method outperforms existing techniques as the number of variables increases.

Abstract

We present a two-dimensional delta symbol method that facilitates a version of the Kloosterman refinement of the circle method, addressing a question posed by Heath-Brown. As an application, we establish the asymptotic formula for the number of integral points on a non-singular intersection of two integral quadratic forms with at least $10$ variables. Assuming the Generalized Lindel\"of Hypothesis, we reduce the number of variables to $9$ by performing a double Kloosterman refinement. A heuristic argument suggests our two-dimensional delta symbol will typically outperform known expressions of this type by an increasing margin as the number of variables grows.