International audienceWe consider the problem of computing univariate polynomial matrices over afield that represent minimal solution bases for a general interpolationproblem, some forms of which are the vector M-Pad\'e approximation problem in[Van Barel and Bultheel, Numerical Algorithms 3, 1992] and the rationalinterpolation problem in [Beckermann and Labahn, SIAM J. Matrix Anal. Appl. 22,2000]. Particular instances of this problem include the bivariate interpolationsteps of Guruswami-Sudan hard-decision and K\"otter-Vardy soft-decisiondecodings of Reed-Solomon codes, the multivariate interpolation step oflist-decoding of folded Reed-Solomon codes, and Hermite-Pad\'e approximation. In the mentioned references, the problem is solved using iterative algorithmsbased on recurrence relations. Here, we discuss a fast, divide-and-conquerversion of this recurrence, taking advantage of fast matrix computations overthe scalars and over the polynomials. This new algorithm is deterministic, andfor computing shifted minimal bases of relations between $m$ vectors of size$\sigma$ it uses $O~( m^{\omega-1} (\sigma + |s|) )$ field operations, where$\omega$ is the exponent of matrix multiplication, and $|s|$ is the sum of theentries of the input shift $s$, with $\min(s) = 0$. This complexity boundimproves in particular on earlier algorithms in the case of bivariateinterpolation for soft decoding, while matching fastest existing algorithms forsimultaneous Hermite-Pad\'e approximation

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We compute minimal bases of solutions for a general interpolation problem, which encompasses Hermite-Padé 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˜(m ω−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-Padé approximation [34] and in the general interpolation case [18], compute non-normalized bases. Since for arbitrary shifts such bases may have size Θ(m 2 σ), the cost bound O˜(m ω−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(mσ). 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.