Arne Storjohann

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A simple algorithm for lattice reduction of polynomial matrices is described and analysed. The algorithm is adapted and applied to various tasks, including rank profile and determinant computation, transformation to Hermite and Popov canonical form, polynomial linear system solving and short vector computation. © 2003 Elsevier Science Ltd. All rights(More)
We reduce the problem of computing the rank and a null-space basis of a univariate polynomial matrix to polynomial matrix multiplication. For an input <i>n</i> x <i>n</i> matrix of degree, <i>d</i> over a field <b>K</b> we give a rank and nullspace algorithm using about the same number of operations as for multiplying two matrices of dimension, <i>n</i> and(More)
This paper presents a new algorithm for computing the Hermite normal form H of an A c Z “m of rank m together with a unimodular pre-multiplier matrix U such that UA = H. Our algorithm requires O-(m ‘-lnM(mlog[lAl\)) bit oper@ions to produce both H and U. Here, IIAII = max,j lAij [, M(t) bit operations are sufficient to multiply two (t] -bit integers, and 0(More)
A sinq)lo randoruixed algorithnl is given for fillding an integer solubion to a s:-stclu of linear Di01)lmnt,ine equations. Givcu as input a s?;stcrrl which admits an int.cger solul.ion, the idgorithru can 1~ used t,o find such a. solution wit,h probabilit~~ ilt least l/2. The running time (nunlber of bit operations) is esscntiall~ cubic in the dimemion of(More)
When s = 1 and N = ‖n‖ − 1 this is the classical Hermite Padé approximation problem. Here we allow N to be arbitrary. We describe algorithms for computing an order N genset of type n: a matrix V ∈ k[x]∗×m such that every row of V is a solution to (1) and every solution P of (1) can be expressed as a k[x]-linear combination of the rows of V . Ideally, V will(More)