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    Fast algorithm for border bases of Artinian Gorenstein algebras

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    Given a multi-index sequence σ\sigma, we present a new efficient algorithm to compute generators of the linear recurrence relations between the terms of σ\sigma. We transform this problem into an algebraic one, by identifying multi-index sequences, multivariate formal power series and linear functionals on the ring of multivariate polynomials. In this setting, the recurrence relations are the elements of the kerne lII\sigma of the Hankel operator $H$\sigma associated to σ\sigma. We describe the correspondence between multi-index sequences with a Hankel operator of finite rank and Artinian Gorenstein Algebras. We show how the algebraic structure of the Artinian Gorenstein algebra AA\sigmaassociatedtothesequence associated to the sequence \sigma yields the structure of the terms $\sigma\alphaforall for all α\alpha ∈\in N n.Thisstructureisexplicitlygivenbyaborderbasisof. This structure is explicitly given by a border basis of Aσ\sigma,whichispresentedasaquotientofthepolynomialring, which is presented as a quotient of the polynomial ring K[x 1 ,. .. , xn]bythekernel] by the kernel Iσ\sigmaoftheHankeloperator of the Hankel operator Hσ\sigma.Thealgorithmprovidesgeneratorsof. The algorithm provides generators of Iσ\sigmaconstitutingaborderbasis,pairwiseorthogonalbasesof constituting a border basis, pairwise orthogonal bases of Aσ\sigma$ and the tables of multiplication by the variables in these bases. It is an extension of Berlekamp-Massey-Sakata (BMS) algorithm, with improved complexity bounds. We present applications of the method to different problems such as the decomposition of functions into weighted sums of exponential functions, sparse interpolation, fast decoding of algebraic codes, computing the vanishing ideal of points, and tensor decomposition. Some benchmarks illustrate the practical behavior of the algorithm
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