Fraction-free row reduction of matrices of Ore polynomials

Popov Form Computation for Matrices of Ore Polynomials

“…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.…”

“…We refer to [2,7] and the references therein for background on linear algebra with matrices of Ore polynomials. Here, we only mention that for matrices over Ore rings the notions of rank and (non)-singularity make sense: in particular, performing row or column operations on a matrix will not change its rank.…”

“…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.…”

“…A common thread in the algorithms mentioned in the previous paragraph is that they either explicitly [7] or implicitly [2] consider a linearization over F of the input matrix: this allows to obtain bounds on the size of intermediate expressions. Although in a slightly different context, we remark that a linear algebra point for normal form computation can be found for the commutative case already in [13, Sections 6.7.1 and 6.72].…”

“…We then perform a structured fraction-free Gaussian elimination to recover the Popov form. The cost of our algorithm is O(n ω+2 d 3 M(n 2 de)) operations from k. Here, ω is an exponent for matrix multiplication, and M is a multiplication time: two polynomials from k[z] of degree strictly less than t can be multiplied in M(t) operations from k. Assuming ω = 3 and a pseudo-linear multiplication time, and ignoring logarithmic factors, the cost of our algorithm is then on the order of n 7 d 4 e operations from k. For comparison, the fraction-free algorithm supporting [2,Corollary 7.7] requires on the order of n 9 d 4 e 2 operations from k to produce a row reduced form of A, while the algorithm in [5, Theorem 6.2] requires on the order of n 8 d 4 e + n 7 d 3 e 2 operations from k. Now consider the case k = Q. Like before, we assume our input matrix is over Z[z][∂; σ, δ].…”