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

2022/03/08 by Csahók, Tímea, Kutas, Péter, Montessinos, Mickaël +1
#FOS: Mathematics #Number Theory (math.NT) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2203.04068

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 M₂(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.

Related

• Explicit equivalence of quadratic forms over \mathbbF_q(t)
• Splitting quaternion algebras over quadratic number fields
• Quaternion quadratic equations in characteristic 2
• Zeros and factorizations of quaternion polynomials: the algorithmic approach
• Quaternions and universal quadratic forms over number fields