A two-dimensional delta symbol method and its application to pairs of quadratic forms
Number Theory
2026-04-30 v2
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 variables. Assuming the Generalized Lindel\"of Hypothesis, we reduce the number of variables to 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.
Cite
@article{arxiv.2411.11355,
title = {A two-dimensional delta symbol method and its application to pairs of quadratic forms},
author = {Junxian Li and Simon L. Rydin Myerson and Pankaj Vishe},
journal= {arXiv preprint arXiv:2411.11355},
year = {2026}
}
Comments
74 Pages, exposition and structure clarified, correct factor of 2 in p_{1,q}