### Abstract

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

Language | English |
---|---|

Pages | 33-63 |

Number of pages | 31 |

Journal | Finite Fields and their Applications |

Volume | 55 |

DOIs | |

Publication status | Published - Jan 1 2019 |

### Fingerprint

### Keywords

- Function field
- Polynomial time algorithm
- Quadratic forms

### ASJC Scopus subject areas

- Theoretical Computer Science
- Algebra and Number Theory
- Engineering(all)
- Applied Mathematics

### Cite this

_{q}(t).

*Finite Fields and their Applications*,

*55*, 33-63. https://doi.org/10.1016/j.ffa.2018.09.003

**Explicit equivalence of quadratic forms over F _{q}(t).** / Ivanyos, Gábor; Kutas, Péter; Rónyai, L.

Research output: Contribution to journal › Article

_{q}(t)'

*Finite Fields and their Applications*, vol. 55, pp. 33-63. https://doi.org/10.1016/j.ffa.2018.09.003

_{q}(t). Finite Fields and their Applications. 2019 Jan 1;55:33-63. https://doi.org/10.1016/j.ffa.2018.09.003

}

TY - JOUR

T1 - Explicit equivalence of quadratic forms over Fq(t)

AU - Ivanyos, Gábor

AU - Kutas, Péter

AU - Rónyai, L.

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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).

AB - 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).

KW - Function field

KW - Polynomial time algorithm

KW - Quadratic forms

UR - http://www.scopus.com/inward/record.url?scp=85053466993&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85053466993&partnerID=8YFLogxK

U2 - 10.1016/j.ffa.2018.09.003

DO - 10.1016/j.ffa.2018.09.003

M3 - Article

VL - 55

SP - 33

EP - 63

JO - Finite Fields and Their Applications

T2 - Finite Fields and Their Applications

JF - Finite Fields and Their Applications

SN - 1071-5797

ER -