# An upper bound on the number of perfect quadratic forms

## TL;DR

This paper improves the upper bound on the number of non-similar perfect quadratic forms in d dimensions from exponential in d^3 log d to exponential in d^2 log d using a volumetric approach based on Voronoi's reduction theory.

## Contribution

The paper presents a tighter upper bound on the count of perfect quadratic forms, refining previous exponential estimates with a volumetric argument.

## Key findings

- Upper bound improved to e^{O(d^2 log d)}
- Uses volumetric argument based on Voronoi's reduction theory
- Refines previous exponential bounds on p_d

## Abstract

In a recent preprint on arXiv Roland Bacher showed that the number $p_d$ of non-similar perfect $d$-dimensional quadratic forms satisfies $e^{\Omega(d)} < p_d < e^{O(d^3\log(d))}$. We improve the upper bound to $e^{O(d^2\log(d))}$ by a volumetric argument based on Voronoi's first reduction theory.