Explicit equivalence of quadratic forms over Fq(t)

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Abstract

We propose a randomized polynomial time algorithm for computing non-trivial zeros of quadratic forms in 4 or more variables over Fq(t), where Fq 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 Fq(t).

MSC

68W30
11E12
11E20

Keywords

Quadratic forms
Function field
Polynomial time algorithm

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