Modular Computation for Matrices of Ore Polynomials

Abstract

We give a modular algorithm to perform row reduction of a matrix of Ore polynomials with coefficients in Z [t]. Both the transformation matrix and the transformed matrix are computed. The algorithm can be used for finding the rank and left nullspace of such matrices. In the special case of shift polynomials, we obtain algorithms for computing a weak Popov form and for computing a greatest common right divisor (GCRD) and a least common left multiple (LCLM) of matrices of shift polynomials. Our algorithms improve on existing fraction-free algorithms and can be viewed as generalizations of the work of Li and Nemes on GCRDs and LCLMs of Ore polynomials. We define lucky homomorphisms, determine the appropriate normalization, as well as bound the number of homomorphic images required. Our algorithm is output-sensitive, such that the number of homomorphic images required depends on the size of the output. Furthermore, there is no need to verify the result by trial division or multiplication. When our algorithm is used to compute a GCRD and a LCLM of shift polynomials, we obtain a new output-sensitive modular algorithm.

Popov Form Computation for Matrices of Ore Polynomials

“…Our algorithm is faster than the row reduction algorithm of [5], which however handles the important case of nonsquare and singular inputs which we do not. Our approach still works for rectangular and rank deficient matrices (of any dimensions) but the complexity could be much higher.…”

“…The modular algorithm supporting [17,Theorem 6.3] requires about n 10 d 5 e log β + n 9 d 4 e 2 log β bit operations. On the one hand, we point out that the algorithms of [2,5] solve a considerably more general problem than we do in this paper: they can be applied to input matrices of arbitrary shape and rank and thus compute the rank of the input matrix as well as a left nullspace. Although we hope to consider the rank deficient case in the future, our analysis currently assumes the input matrix is non-singular.…”

“…Beckermann, Cheng and Labahn [2] and Cheng and Labahn [5] compute row reduced bases using an "order basis" approach as known for matrices over F [x], and taking care of coefficient growth. Davies, Cheng and Labahn [17] show how computing the Popov form can be reduced to nullspace computation, a problem for which effective fraction-free techniques exist.…”

“…Although we hope to consider the rank deficient case in the future, our analysis currently assumes the input matrix is non-singular. On the other hand, the algorithms in [2,5] only produce a row reduced form of A and not the canonical Popov forms.…”

Computing diagonal form and Jacobson normal form of a matrix using Gröbner bases

“…Implementations of Jacobson normal form. To the best of our knowledge, Jacobson normal form algorithm has been implemented in Maple by Culianez and Quadrat [11], by Robertz et al [4,8], by Middeke [28] and by Cheng et al [3,6,12].…”