Fast Computation of Shifted Popov Forms of Polynomial Matrices via
Systems of Modular Polynomial Equations
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by
Vincent Neiger
2016
Abstract
We give a Las Vegas algorithm which computes the shifted Popov form of an m
× m nonsingular polynomial matrix of degree d in expected
O(m^ω d) field operations, where ω is the
exponent of matrix multiplication and O(·)
indicates that logarithmic factors are omitted. This is the first algorithm in
O(m^ω d) for shifted row reduction with arbitrary
shifts.
Using partial linearization, we reduce the problem to the case d <σ/m where σ is the generic determinant bound, with σ
/ m bounded from above by both the average row degree and the average column
degree of the matrix. The cost above becomes O(m^ωσ/m ), improving upon the cost of the fastest previously
known algorithm for row reduction, which is deterministic.
Our algorithm first builds a system of modular equations whose solution set
is the row space of the input matrix, and then finds the basis in shifted Popov
form of this set. We give a deterministic algorithm for this second step
supporting arbitrary moduli in O(m^ω-1σ)
field operations, where m is the number of unknowns and σ is the sum
of the degrees of the moduli. This extends previous results with the same cost
bound in the specific cases of order basis computation and M-Padé
approximation, in which the moduli are products of known linear factors.
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