Solving quadratic forms in restricted variables with the circle method

TL;DR

This paper develops conditions under which the circle method can be effectively used to count solutions to quadratic forms with restricted variables, focusing on forms with many mixed terms and weighted solutions.

Contribution

It introduces new criteria for applying the circle method to quadratic forms with restrictions and weights, expanding its applicability to more complex forms.

Findings

Established conditions for the circle method to count solutions

Demonstrated effectiveness for forms with many mixed terms

Provided bounds on solutions of bounded height

Abstract

Let $f(\mathbf x)$ be a non-singular quadratic form with sufficiently many mixed terms and $t$ an integer. For a sequence of weights $\mathcal A$ we study the number of weighted solutions to $f(\mathbf x) = t$. In particular, we give conditions on both $\mathcal A$ and $f$ such that we can use the circle method to count such solutions of bounded height.