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

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 ${\mathbb{F}}_q(t)$.

27 pages, revised version, accepted for publication in Finite Fields and Their Applications

References in corpus (1)

• Computing explicit isomorphisms with full matrix algebras over ${\mathbb{F}}_q(x)$