Finding nontrivial zeros of quadratic forms over rational function fields of characteristic 2

TL;DR

This paper introduces polynomial-time algorithms for finding nontrivial zeros of quadratic forms over rational function fields of characteristic 2, with practical implementation demonstrating effectiveness.

Contribution

It presents the first efficient algorithms for this problem and applies them to identify quadratic subfields and zero divisors in matrix algebras over division algebras.

Findings

Algorithms run in polynomial time

Successful implementation in MAGMA

Practical applicability demonstrated

Abstract

We propose polynomial-time algorithms for finding nontrivial zeros of quadratic forms with four variables over rational function fields of characteristic 2. We apply these results to find prescribed quadratic subfields of quaternion division division algebras and zero divisors in $M_2(D)$, the full matrix algebra over a division algebra, given by structure constants. We also provide an implementation of our results in MAGMA which shows that the algorithms are truly practical.