Fast algorithm for border bases of Artinian Gorenstein algebras

Given a multi-index sequence $$\sigma$$, we present a new efficient algorithm to compute generators of the linear recurrence relations between the terms of $$\sigma$$. 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 l$I$\sigma of the Hankel operator $H$\sigma associated to $$\sigma$$. 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 $A$\sigma$$associated to the sequence$$\sigma yields the structure of the terms $\sigma\alpha$for all$$\alpha$ $\in$ N n$. This structure is explicitly given by a border basis of$A$\sigma$$, which is presented as a quotient of the polynomial ring$K[x 1 ,. .. , xn$] by the kernel$I$\sigma$$of the Hankel operator$H$\sigma$$. The algorithm provides generators of$I$\sigma$$constituting a border basis, pairwise orthogonal bases of$A$\sigma$$ 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

Algorithms for zero-dimensional ideals using linear recurrent sequences

Inspired by Faug\`ere and Mou's sparse FGLM algorithm, we show how using linear recurrent multi-dimensional sequences can allow one to perform operations such as the primary decomposition of an ideal, by computing the annihilator of one or several such sequences.Comment: LNCS, Computer Algebra in Scientific Computing CASC 201

Computing real radicals by moment optimization

We present a new algorithm for computing the real radical of an ideal and, more generally, the-radical of, which is based on convex moment optimization. A truncated positive generic linear functional vanishing on the generators of is computed solving a Moment Optimization Problem (MOP). We show that, for a large enough degree of truncation, the annihilator of generates the real radical of. We give an effective, general stopping criterion on the degree to detect when the prime ideals lying over the annihilator are real and compute the real radical as the intersection of real prime ideals lying over. The method involves several ingredients, that exploit the properties of generic positive moment sequences. A new efficient algorithm is proposed to compute a graded basis of the annihilator of a truncated positive linear functional. We propose a new algorithm to check that an irreducible decomposition of an algebraic variety is real, using a generic real projection to reduce to the hypersurface case. There we apply the Sign Changing Criterion, effectively performed with an exact MOP. Finally we illustrate our approach in some examples.Comment: ISSAC 2021 - 46th International Symposium on Symbolic and Algebraic Computation, Jul 2021, Saint-P{\'e}tersbourg, Russi

Generalised power series determined by linear recurrence relations

In 1882, Kronecker established that a given univariate formal Laurent series over a field can be expressed as a fraction of two univariate polynomials if and only if the coefficients of the series satisfy a linear recurrence relation. We introduce the notion of generalised linear recurrence relations for power series with exponents in an arbitrary ordered abelian group, and generalise Kronecker's original result. In particular, we obtain criteria for determining whether a multivariate formal Laurent series lies in the fraction field of the corresponding polynomial ring. Moreover, we study distinguished algebraic substructures of a power series field, which are determined by generalised linear recurrence relations. In particular, we identify generalised linear recurrence relations that determine power series fields satisfying additional properties which are essential for the study of their automorphism groups.Comment: 33 pages, submitte

Polynomial-Division-Based Algorithms for Computing Linear Recurrence Relations

Sparse polynomial interpolation, sparse linear system solving or modular rational reconstruction are fundamental problems in Computer Algebra. They come down to computing linear recurrence relations of a sequence with the Berlekamp-Massey algorithm. Likewise, sparse multivariate polynomial interpolation and multidimensional cyclic code decoding require guessing linear recurrence relations of a multivariate sequence.Several algorithms solve this problem. The so-called Berlekamp-Massey-Sakata algorithm (1988) uses polynomial additions and shifts by a monomial. The Scalar-FGLM algorithm (2015) relies on linear algebra operations on a multi-Hankel matrix, a multivariate generalization of a Hankel matrix. The Artinian Gorenstein border basis algorithm (2017) uses a Gram-Schmidt process.We propose a new algorithm for computing the Gr{\"o}bner basis of the ideal of relations of a sequence based solely on multivariate polynomial arithmetic. This algorithm allows us to both revisit the Berlekamp-Massey-Sakata algorithm through the use of polynomial divisions and to completely revise the Scalar-FGLM algorithm without linear algebra operations.A key observation in the design of this algorithm is to work on the mirror of the truncated generating series allowing us to use polynomial arithmetic modulo a monomial ideal. It appears to have some similarities with Pad{\'e} approximants of this mirror polynomial.As an addition from the paper published at the ISSAC conferance, we give an adaptive variant of this algorithm taking into account the shape of the final Gr{\"o}bner basis gradually as it is discovered. The main advantage of this algorithm is that its complexity in terms of operations and sequence queries only depends on the output Gr{\"o}bner basis.All these algorithms have been implemented in Maple and we report on our comparisons

Symmetry in Multivariate Ideal Interpolation

An interpolation problem is defined by a set of linear forms on the (multivariate) polynomial ring and values to be achieved by an interpolant. For Lagrange interpolation the linear forms consist of evaluations at some nodes,while Hermite interpolation also considers the values of successive derivatives. Both are examples of ideal interpolation in that the kernels of the linear forms intersect into an ideal. For an ideal interpolation problem with symmetry, we address the simultaneous computation of a symmetry adapted basis of the least interpolation space and the symmetry adapted H-basis of the ideal. Beside its manifest presence in the output, symmetry is exploited computationally at all stages of the algorithm. For an ideal invariant, under a group action, defined by a Groebner basis, the algorithm allows to obtain a symmetry adapted basis of the quotient and of the generators. We shall also note how it applies surprisingly but straightforwardly to compute fundamental invariants and equivariants of a reflection group

Algorithms for Linearly Recurrent Sequences of Truncated Polynomials

Linear recurrent sequences are those whose elements are defined as linear combinations of preceding elements, and finding recurrence relations is a fundamental problem in computer algebra. In this paper, we focus on sequences whose elements are vectors over the ring A = K[x]/ of truncated polynomials. Finding the ideal of their recurrence relations has applications such as the computation of minimal polynomials and determinants of sparse matrices over A. We present three methods for finding this ideal: a Berlekamp-Massey-like approach due to Kurakin, one which computes the kernel of some block-Hankel matrix over A via a minimal approximant basis, and one based on bivariate Pade approximation. We propose complexity improvements for the first two methods, respectively by avoiding the computation of redundant relations and by exploiting the Hankel structure to compress the approximation problem. Then we confirm these improvements empirically through a C++ implementation, and we discuss the above-mentioned applications