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

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

[1]  Johanna Weiss,et al.  Arithmetique Des Algebres De Quaternions , 2016 .

[2]  E. Landau,et al.  Über die Primfunktionen in einer arithmetischen Progression , 1919 .

[3]  Gábor Ivanyos,et al.  Lattice basis reduction for indefinite forms and an application , 1996, Discret. Math..

[4]  Herbert Rauter Über die Darstellbarkeit durch quadratische Formen im Körper der rationalen Funktionen einer Unbestimmten über dem Restklassenkörper mod p , 1926 .

[5]  Christiaan E. van de Woestijne,et al.  Deterministic equation solving over finite fields , 2005, ISSAC.

[6]  Mireille Car,et al.  Répartition modulo 1 dans un corps de séries formelles sur un corps fini , 1995 .

[7]  Lajos Rónyai,et al.  Splitting full matrix algebras over algebraic number fields , 2011, ArXiv.

[8]  Lajos Rónyai,et al.  Simple algebras are difficult , 1987, STOC.

[9]  David Y. Y. Yun,et al.  On square-free decomposition algorithms , 1976, SYMSAC '76.

[10]  Tsit Yuen Lam,et al.  Introduction To Quadratic Forms Over Fields , 2004 .

[11]  John Cremona,et al.  Efficient solution of rational conics , 2003, Math. Comput..

[12]  Lajos Rónyai,et al.  Factoring polynomials over finite fields , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[13]  Gove Effinger,et al.  Additive Number Theory of Polynomials Over a Finite Field , 1991 .

[14]  Pierre Castel,et al.  Un algorithme de résolution des équations quadratiques en dimension 5 sans factorisation , 2011 .

[15]  Mark van Hoeij,et al.  Solving conics over function fields , 2006 .

[16]  Daqing Wan,et al.  Generators and irreducible polynomials over finite fields , 1997, Math. Comput..

[17]  D. Hoffmann,et al.  INTRODUCTION TO QUADRATIC FORMS OVER FIELDS (Graduate Studies in Mathematics 67) By T. Y. L AM : 550 pp., US$79.00, ISBN 0-8218-1095-2 (American Mathematical Society, Providence, RI, 2005) , 2005 .

[18]  Larry J. Gerstein Basic quadratic forms , 2008 .

[19]  J. Neukirch Algebraic Number Theory , 1999 .