Popov Form Computation for Matrices of Ore Polynomials

Abstract

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 orthogonality 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.

Computing Valuations of the Dieudonné Determinants

“…Note again that the cost is in operations over and calls of . [17], the Hermite normal form [12], the Popov normal form [15] and their weaker form called a row-reduced form [1,2]. One can use these algorithms to calculate deg Det since it is immediately obtained from the canonical forms of .…”

Fast Root Finding for Interpolation-Based Decoding of Interleaved Gabidulin Codes

“…Ore rings over infinite rings such as Q(t) have interest in time-dependent systems and differential equations, and since the late 1990s' the computer algebra community has developed algorithms for minimal approximants and row reductions of matrices over Ore rings, see e.g. [28], [29] and references therein, but with a focus on handling the coefficient growth of infinite rings. For the finite case, in particular L q m[x], faster methods are available, e.g.…”