Computing the $p$-adic Canonical Quadratic Form in Polynomial Time

TL;DR

This paper introduces a randomized polynomial time algorithm for computing the $p$-adic canonical form of an integral quadratic form, facilitating efficient classification over local fields.

Contribution

The paper presents the first polynomial time algorithm to compute the $p$-adic canonical form of quadratic forms, improving computational methods in number theory.

Findings

Algorithm runs in polynomial time

Successfully computes $p$-adic canonical forms

Applicable to general integral quadratic forms

Abstract

An $n$-ary integral quadratic form is a formal expression $Q(x_1,..,x_n)=\sum_{1\leq i,j\leq n}a_{ij}x_ix_j$ in $n$-variables $x_1,...,x_n$, where $a_{ij}=a_{ji} \in \mathbb{Z}$. We present a randomized polynomial time algorithm that given a quadratic form $Q(x_1,...,x_n)$, a prime $p$, and a positive integer $k$ outputs a $\mathtt{U} \in \text{GL}_n(\mathbb{Z}/p^k\mathbb{Z})$ such that $\mathtt{U}$ transforms $Q$ to its $p$-adic canonical form.