Computing Explicit Isomorphisms with Full Matrix Algebras over $$\mathbb {F}_q(x)$$Fq(x)

We propose a polynomial time f-algorithm (a deterministic algorithm which uses an oracle for factoring univariate polynomials over $$\mathbb {F}_q$$Fq) for computing an isomorphism (if there is any) of a finite-dimensional $$\mathbb {F}_q(x)$$Fq(x)-algebra $$\mathcal{A}$$A given by structure constants with the algebra of n by n matrices with entries from $$\mathbb {F}_q(x)$$Fq(x). The method is based on computing a finite $$\mathbb {F}_q$$Fq-subalgebra of $$\mathcal{A}$$A which is the intersection of a maximal $$\mathbb {F}_q[x]$$Fq[x]-order and a maximal R-order, where R is the subring of $$\mathbb {F}_q(x)$$Fq(x) consisting of fractions of polynomials with denominator having degree not less than that of the numerator.

[2]  Lajos Rónyai,et al.  Decomposition of algebras overFq(X1,...,Xm) , 1994, Applicable Algebra in Engineering, Communication and Computing.

[3]  Miklós Ajtai,et al.  The shortest vector problem in L2 is NP-hard for randomized reductions (extended abstract) , 1998, STOC '98.

[4]  F. J. Lobillo,et al.  Factoring Ore polynomials over $\mathbb{F}_q(t)$ is difficult , 2015, 1505.07252.

[5]  Lajos Rónyai,et al.  Computing the Structure of Finite Algebras , 1990, J. Symb. Comput..

[6]  M. Ajtai.  The shortest vector problem in L2 is NP-hard for randomized reductions (extended abstract) , 1998, STOC '98.

[7]  J. Cremona,et al.  Explicit n-descent on elliptic curves, I. Algebra , 2006, math/0606580.

[8]  Y. Drozd,et al.  Finite dimensional algebras , 1994 .

[9]  John Cremona,et al.  Explicit n-descent on elliptic curves III. Algorithms , 2011, Math. Comput..

[10]  Lajos Rónyai,et al.  Polynomial time solutions of some problems of computational algebra , 1985, STOC '85.

[11]  D. Cantor,et al.  A new algorithm for factoring polynomials over finite fields , 1981 .

[12]  Mark Giesbrecht,et al.  Factoring and decomposing ore polynomials over Fq(t) , 2003, ISSAC '03.

[13]  G. Ivanyos.  Algorithms for algebras over global fields , 1996 .

[14]  Lajos Rónyai,et al.  Computing Levi Decompositions in Lie algebras , 1997, Applicable Algebra in Engineering, Communication and Computing.

[15]  Sachar Paulus,et al.  Lattice Basis Reduction in Function Fields , 1998, ANTS.

[16]  Lajos Rónyai,et al.  Finding maximal orders in semisimple algebras over Q , 1993, computational complexity.

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

[18]  Marek Karpinski,et al.  Trading GRH for algebra: Algorithms for factoring polynomials and related structures , 2008, Math. Comput..

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

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

[21]  Arjen K. Lenstra,et al.  Factoring multivariate polynomials over finite fields , 1983, J. Comput. Syst. Sci..

[22]  J. Cremona,et al.  Explicit n-descent on elliptic curves, II. Geometry , 2006, math/0611606.