Paper Info
Title
Fast computation of minimal interpolation bases in Popov form for arbitrary shifts.
Abstract
We compute minimal bases of solutions for a general interpolation problem, which encompasses Hermite-Pade approximation and constrained multivariate interpolation, and has applications in coding theory and security. This problem asks to find univariate polynomial relations between m vectors of size σ; these relations should have small degree with respect to an input degree shift. For an arbitrary shift, we propose an algorithm for the computation of an interpolation basis in shifted Popov normal form with a cost of O~(mω-1 σ) field operations, where ω is the exponent of matrix multiplication and the notation O~(·) indicates that logarithmic terms are omitted. Earlier works, in the case of Hermite-Pade approximation and in the general interpolation case, compute non-normalized bases. Since for arbitrary shifts such bases may have size Θ(m2 σ), the cost bound O~(mω-1 σ) was feasible only with restrictive assumptions on the shift that ensure small output sizes. The question of handling arbitrary shifts with the same complexity bound was left open. To obtain the target cost for any shift, we strengthen the properties of the output bases, and of those obtained during the course of the algorithm: all the bases are computed in shifted Popov form, whose size is always O(m σ). Then, we design a divide-and-conquer scheme. We recursively reduce the initial interpolation problem to sub-problems with more convenient shifts by first computing information on the degrees of the intermediate bases.
Year
DOI
Venue
2016
10.1145/2930889.2930928
ISSAC
Keywords
DocType
Volume
M-Pade approximation, Hermite-Pade approximation, order basis, polynomial matrix, shifted Popov form
Conference
abs/1602.00651
Citations 
PageRank 
References 
8
0.47
22
Authors
4
Name
Order
Citations
PageRank
Claude-Pierre Jeannerod122422.05
Vincent Neiger2397.22
Éric Schost371258.00
Gilles Villard456548.04