Section:
New Results
Fast algorithm for border bases of Artinian Gorenstein algebras
Participant :
Bernard Mourrain.
Given a multi-index sequence , we present in [23] a new
efficient algorithm to compute generators of the linear recurrence
relations between the terms of . 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 kernel of the Hankel operator
associated to . 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 associated to the
sequence yields the structure of the terms
for all . This structure is explicitly given by a border
basis of , which is presented as a quotient of the polynomial ring
by the kernel of the Hankel
operator . The algorithm provides generators of
constituting a border basis, pairwise orthogonal bases of
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.