# A Finiteness theorem for positive definite strictly $n$-regular quadratic forms

## TL;DR

This paper proves that for any fixed number of variables n ≥ 2, there are only finitely many positive definite strictly n-regular quadratic forms in n+4 variables, extending previous finiteness results.

## Contribution

It establishes a finiteness theorem for positive definite strictly n-regular quadratic forms in n+4 variables, generalizing recent results for quaternary forms.

## Key findings

- Finiteness of similarity classes for strictly n-regular forms in n+4 variables
- Extension of finiteness results from quaternary to higher dimensions
- Provides a classification framework for these quadratic forms

## Abstract

An integral quadratic form is called strictly $n$-regular if it primitively represents all quadratic forms in $n$ variables that are primitively represented by its genus. For any $n \geq 2$, it will be shown that there are only finitely many similarity classes of positive definite strictly $n$-regular integral quadratic forms in $n + 4$ variables. This extends the recent finiteness results for strictly regular quaternary quadratic forms by Earnest-Kim-Meyer (2014).