Forms representing forms: The definite case

TL;DR

This paper develops an asymptotic formula for counting representations of one positive definite form by another using the circle method, with applications to quadratic forms and linear spaces in hypersurfaces.

Contribution

It extends the circle method to general positive definite forms, providing new results on representations and geometric configurations.

Findings

Established an asymptotic formula for form representations.

Supersedes previous quadratic form results by Dietmann and Harvey.

Applied to count primitive linear spaces in hypersurfaces.

Abstract

Let $\psi$ and $F$ be positive definite forms with integral coefficients of equal degree. Using the circle method, we establish an asymptotic formula for the number of identical representations of $\psi$ by $F$, provided $\psi$ is everywhere locally representable and the number of variables of $F$ is large enough. In the quadratic case this supersedes a recent result due to Dietmann and Harvey. Another application addresses the number of primitive linear spaces contained in a hypersurface.