Linear Algebra for Computing Gröbner Bases of Linear Recursive Multidimensional Sequences

International audienceSakata generalized the Berlekamp -- Massey algorithm to $n$ dimensions in~1988. The Berlekamp -- Massey -- Sakata (BMS)algorithm can be used for finding a Gröbner basis of a $0$-dimensionalideal of relations verified by a table. We investigate this problem usinglinear algebra techniques, with motivations such as accelerating change ofbasis algorithms (FGLM) or improving their complexity.We first define and characterize multidimensional linear recursive sequencesfor $0$-dimensional ideals.Under genericity assumptions, we propose a randomized preprocessing of thetable that corresponds to performing a linear change of coordinates on thepolynomials associated with the linear recurrences. This technique thenessentially reduces our problem to using the efficient $1$-dimensional Berlekamp -- Massey (BM)algorithm.However, the number of probes to the table in this scheme may be elevated.We thus consider the table in the \emph{black-box} model: we assume probing thetable is expensive and we minimize the number of probes to the table in ourcomplexity model.We produce an FGLM-like algorithm for finding the relations in thetable, which lets us use linear algebra techniques. Under some additionalassumptions, we make this algorithm adaptive and reduce further the numberof table probes.This number can be estimated by counting the number of distinct elements in amulti-Hankel matrix (a multivariate generalization of Hankel matrices); we canrelate this quantity with the \emph{geometry} of the final staircase. Hence,in favorable cases such as convex ones, the complexity is essentially linear inthe size of the output. Finally, when using the \textsc{lex} ordering, we canmake use of fast structured linear algebra similarly to the Hankelinterpretation of Berlekamp -- Massey

Linear Algebra for Computing Gröbner Bases of Linear Recursive Multidimensional Sequences

Special issue on the conference ISSAC 2015: Symbolic computation and computer algebraInternational audienceThe so-called Berlekamp~-- Massey~-- Sakata algorithmcomputes a Gröbner basis of a $0$-dimensional ideal of relations satisfied by an inputtable. It extends the Berlekamp~-- Massey algorithmto $n$-dimensional tables, for $n>1$.We investigate this problem and design several algorithms forcomputing such a Gröbner basis of an ideal of relations using linearalgebra techniques.The first one performs a lot of table queries andis analogous to a change of variables on the ideal of relations.As each query to the table can be expensive,we design a second algorithmrequiring fewer queries, in general.This \textsc{FGLM}-like algorithm allows us to compute the relations of thetable by extracting a full rank submatrix of a \emph{multi-Hankel}matrix (a multivariate generalization of Hankel matrices).Under someadditional assumptions, we make a third, adaptive, algorithm and reducefurther the number of table queries.Then, we relate the number of queries ofthis third algorithm to the\emph{geometry} of the final staircase and we show that it isessentially linear in the size of the output when the staircase is convex.As a direct application to this, we decode $n$-cyclic codes, ageneralization in dimension $n$ of Reed Solomon codes. We show that the multi-Hankelmatrices are heavily structured when using the \textsc{LEX} orderingand that we can speed up the computations using fast algorithms forquasi-Hankel matrices.Finally, we designalgorithms for computing the generating series of a linear recursivetable

A polynomial-division-based algorithm for computing linear recurrence relations

International audienceSparse 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 multidi-mensional 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 Gröbner 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 Padé approximants of this mirror polynomial. Finally, we give a partial solution to the transformation of this algorithm into an adaptive one

Gröbner Basis over Semigroup Algebras: Algorithms and Applications for Sparse Polynomial Systems

International audienceGröbner bases is one the most powerful tools in algorithmic non-linear algebra. Their computation is an intrinsically hard problem with a complexity at least single exponential in the number of variables. However, in most of the cases, the polynomial systems coming from applications have some kind of structure. For example , several problems in computer-aided design, robotics, vision, biology , kinematics, cryptography, and optimization involve sparse systems where the input polynomials have a few non-zero terms. Our approach to exploit sparsity is to embed the systems in a semigroup algebra and to compute Gröbner bases over this algebra. Up to now, the algorithms that follow this approach benefit from the sparsity only in the case where all the polynomials have the same sparsity structure, that is the same Newton polytope. We introduce the first algorithm that overcomes this restriction. Under regularity assumptions, it performs no redundant computations. Further, we extend this algorithm to compute Gröbner basis in the standard algebra and solve sparse polynomials systems over the torus $(C^*)^n$. The complexity of the algorithm depends on the Newton polytopes

Guessing Linear Recurrence Relations of Sequence Tuples and P-recursive Sequences with Linear Algebra

International audienceGiven several $n$-dimensional sequences, we first present an algorithmfor computing the Gröbner basis of their module of linear recurrencerelations.A P-recursive sequence $(u_{\mathbf{i}})_{\mathbf{i}\in\mathbb{N}^n}$satisfies linear recurrence relations with polynomial coefficients in$\mathbf{i}$, as defined by Stanley in 1980. Calling directlythe aforementioned algorithm on the tuple ofsequences $\left((\mathbf{i}^{\mathbf{j}}\,u_{\mathbf{i}})_{\mathbf{i}\in\mathbb{N}^n}\right)_{\mathbf{j}}$for retrieving the relations yields redundant relations.Since the module of relations of aP-recursive sequence also has an extra structure of a $0$-dimensional rightideal of an Ore algebra, we design a more efficient algorithm that takesadvantage of this extra structure forcomputing the relations.Finally, we show how to incorporate Gröbner bases computations in anOre algebra $\mathbb{K}\langle t_1,\ldots,t_n,x_1,\ldots,x_n\rangle$, withcommutators $x_k\,x_{\ell}-x_{\ell}\,x_k=t_k\,t_{\ell}-t_{\ell}\,t_k=t_k\,x_{\ell}-x_{\ell}\,t_k=0$ for $k\neq\ell$ and$t_k\,x_k-x_k\,t_k=x_k$, into the algorithm designed for P-recursivesequences. This allows us to compute faster the Gr\"obner basis of the ideal spanned by the first relations,such as in \textsc{2D}/\textsc{3D}-space walks examples

Ideal Interpolation, H-Bases and Symmetry

International audienceMultivariate Lagrange and Hermite interpolation are examples ofideal interpolation. More generally an ideal interpolation problemis defined by a set of linear forms, on the polynomial ring, whosekernels intersect into an ideal.For an ideal interpolation problem with symmetry, we addressthe simultaneous computation of a symmetry adapted basis of theleast interpolation space and the symmetry adapted H-basis ofthe ideal. Beside its manifest presence in the output, symmetry isexploited computationally at all stages of the algorithm

Algorithms in Intersection Theory in the Plane

This thesis presents an algorithm to find the local structure of intersections of plane curves. More precisely, we address the question of describing the scheme of the quotient ring of a bivariate zero-dimensional ideal $I\subseteq \mathbb K[x,y]$, \textit{i.e.} finding the points (maximal ideals of $\mathbb K[x,y]/I$) and describing the regular functions on those points. A natural way to address this problem is via Gr\"obner bases as they reduce the problem of finding the points to a problem of factorisation, and the sheaf of rings of regular functions can be studied with those bases through the division algorithm and localisation. Let $I\subseteq \mathbb K[x,y]$ be an ideal generated by $\mathcal F$, a subset of $\mathbb A[x,y]$ with $\mathbb A\hookrightarrow\mathbb K$ and $\mathbb K$ a field. We present an algorithm that features a quadratic convergence to find a Gr\"obner basis of $I$ or its primary component at the origin. We introduce an $\mathfrak m$-adic Newton iteration to lift the lexicographic Gr\"obner basis of any finite intersection of zero-dimensional primary components of $I$ if $\mathfrak m\subseteq \mathbb A$ is a \textit{good} maximal ideal. It relies on a structural result about the syzygies in such a basis due to Conca \textit{\&} Valla (2008), from which arises an explicit map between ideals in a stratum (or Gr\"obner cell) and points in the associated moduli space. We also qualify what makes a maximal ideal $\mathfrak m$ suitable for our filtration. When the field $\mathbb K$ is \textit{large enough}, endowed with an Archimedean or ultrametric valuation, and admits a fraction reconstruction algorithm, we use this result to give a complete $\mathfrak m$-adic algorithm to recover $\mathcal G$, the Gr\"obner basis of $I$. We observe that previous results of Lazard that use Hermite normal forms to compute Gr\"obner bases for ideals with two generators can be generalised to a set of $n$ generators. We use this result to obtain a bound on the height of the coefficients of $\mathcal G$ and to control the probability of choosing a \textit{good} maximal ideal $\mathfrak m\subseteq\mathbb A$ to build the $\mathfrak m$-adic expansion of $\mathcal G$. Inspired by Pardue (1994), we also give a constructive proof to characterise a Zariski open set of $\mathrm{GL}_2(\mathbb K)$ (with action on $\mathbb K[x,y]$) that changes coordinates in such a way as to ensure the initial term ideal of a zero-dimensional $I$ becomes Borel-fixed when $|\mathbb K|$ is sufficiently large. This sharpens our analysis to obtain, when $\mathbb A=\mathbb Z$ or $\mathbb A=k[t]$, a complexity less than cubic in terms of the dimension of $\mathbb Q[x,y]/\langle \mathcal G\rangle$ and softly linear in the height of the coefficients of $\mathcal G$. We adapt the resulting method and present the analysis to find the $\langle x,y\rangle$-primary component of $I$. We also discuss the transition towards other primary components via linear mappings, called \emph{untangling} and \emph{tangling}, introduced by van der Hoeven and Lecerf (2017). The two maps form one isomorphism to find points with an isomorphic local structure and, at the origin, bind them. We give a slightly faster tangling algorithm and discuss new applications of these techniques. We show how to extend these ideas to bivariate settings and give a bound on the arithmetic complexity for certain algebras

Algèbre linéaire dédiée pour les algorithmes Scalar-FGLM et Berlekamp-Massey-Sakata

International audienceLe contexte général L'algorithme de Berlekamp-Massey ([1], [11]) a été inventé par Berlekamp pour décoder les codes BCH ([5]), puis Massey a montré qu'il permettait de résoudre le problème de devinette de récurrence linéaire pour les suites à un indice. Il a ensuite été étendu par Sakata ([13]) pour résoudre le même problème pour les suites à plusieurs indices (algorithme de Berlekamp-Massey-Sakata, ou BMS). La solution prend alors la forme d'une base de Gröbner de l'idéal des relations d'une table de valeurs de la suite. Il a enfin été légèrement adapté pour permettre le décodage des codes d'évaluations sur un domaine ordonné ([6]). Récemment, dans [3], Faugère, Berthomieu et Boyer on présenté un al-gorithme, Scalar-FGLM, généralisant la version matricielle de l'algorithme de Berlekamp-Massey pour les suites à plusieurs indices. Cette dernière consiste à résoudre un système linéaire de Hankel de taille d, l'ordre de la récurrence. Ceci est possible en complexité en temps O (M(d) log d), où M(d) est le coût du produit de polynômes de degré d (voir [7]). Dans le cas de Scalar-FGLM, on extrait une sous matrice de rang maximal d'une matrice multi-Hankel puis on résout des systèmes faisant intervenir directement cette matrice. Cela est possible en complexité O (M(d) log d) pour l'ordre lexicographique et dans le cas générique (shape position). Cependant, sans cette hypothèse de généricité on ne sait pas résoudre le système linéaire aussi efficacement. Dans [4], les auteurs de Scalar-FGLM l'étendent aux suites linéaires récu-rentes à coefficients polynomiaux, les suites P-récurrentes. Le problème ouvert de savoir si certaines marches de l'espaces sont P-récurrentes ou non pourrait se voir apporter des réponses en cas de progrès pratiques ou théoriques dans la gestion des matrices générées par Scalar-FGLM. Le problème étudié Un premier objectif du stage était d'obtenir une algèbre linéaire plus rapide en pratique ou en théorie pour Scalar-FGLM et ses dérivés. Un second était de rapprocher les descriptions de Scalar-FGLM et de BMS, puisque ces deux algorithmes ont des sorties équivalentes