Fast Computation of Minimal Interpolation Bases in Popov Form for Arbitrary Shifts

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 sigma; 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-similar to(m(omega-1)sigma) field operations, where omega is the exponent of matrix multiplication and the notation O-similar to(.) indicates that logarithmic terms are omitted. Earlier works, in the case of Hermite-Pade approximation [34] and in the general interpolation case [18], compute non-normalized bases. Since for arbitrary shifts such bases may have size Theta(m(2)sigma), the cost bound O-similar to(m(omega-1)sigma) 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 sigma). 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.