Explicit equivalence of quadratic forms over $\mathbb{F}_q(t)$

TL;DR

This paper introduces a randomized polynomial-time algorithm for finding nontrivial zeros of quadratic forms over function fields, enabling applications like Witt decomposition and isometry computation.

Contribution

It presents a novel randomized algorithm for quadratic forms over $\\mathbb{F}_q(t)$, including methods for Witt decomposition and isometry, with applications to quaternion algebras.

Findings

Efficient algorithm for zeros of quadratic forms in 4+ variables

Application to Witt decomposition and isometry of quadratic forms

Computing zero divisors in quaternion algebras

Abstract

We propose a randomized polynomial time algorithm for computing nontrivial zeros of quadratic forms in 4 or more variables over $\mathbb{F}_q(t)$, where $\mathbb{F}_q$ is a finite field of odd characteristic. The algorithm is based on a suitable splitting of the form into two forms and finding a common value they both represent. We make use of an effective formula for the number of fixed degree irreducible polynomials in a given residue class. We apply our algorithms for computing a Witt decomposition of a quadratic form, for computing an explicit isometry between quadratic forms and finding zero divisors in quaternion algebras over quadratic extensions of $\mathbb{F}_q(t)$.