Popov Form Computation for Matrices of Ore Polynomials

Mohamed Khochtali, Johan Rosenkilde né Nielsen, Arne Storjohann
2017 Proceedings of the 2017 ACM on International Symposium on Symbolic and Algebraic Computation - ISSAC '17  
Let F[∂; σ, δ] be a ring of Ore polynomials over a field. We give a new deterministic algorithm for computing the Popov form P of a non-singular matrix A ∈ F[∂; σ, δ] n×n . Our main focus is to ensure controlled growth in the size of coefficients from F in the case F = k(z), and even k = Q. Our algorithms are based on constructing from A a linear system over F and performing a structured fraction-free Gaussian elimination. The algorithm is output sensitive, with a cost that depends on the
more » ... onality defect of the input matrix: the sum of the row degrees in A minus the sum of the row degrees in P . The resulting bit-complexity for the differential and shift polynomial case over Q(z) improves upon the previous best.
doi:10.1145/3087604.3087650 dblp:conf/issac/KhochtaliRS17 fatcat:feha4hlvdbeffjrcra4zyum2bm