18 research outputs found
On the Reduced Gr\"obner Bases of Blockwise Determinantal Ideals
Blockwise determinantal ideals are those generated by the union of all the
minors of specified sizes in certain blocks of a generic matrix, and they are
the natural generalization of many existing determinantal ideals like the
Schubert and ladder ones. In this paper we establish several criteria to verify
whether the Gr\"obner bases of blockwise determinantal ideals with respect to
(anti-)diagonal term orders are minimal or reduced. In particular, for Schubert
determinantal ideals, while all the elusive minors form the reduced Gr\"obner
bases when the defining permutations are vexillary, in the non-vexillary case
we derive an explicit formula for computing the reduced Gr\"obner basis from
elusive minors which avoids all algebraic operations. The fundamental
properties of being normal and strong for W-characteristic sets and
characteristic pairs, which are heavily connected to the reduced Gr\"obner
bases, of Schubert determinantal ideals are also proven
Sparse FGLM algorithms
International audienceGiven a zero-dimensional ideal I \subset \kx of degree , the transformation of the ordering of its \grobner basis from DRL to LEX is a key step in polynomial system solving and turns out to be the bottleneck of the whole solving process. Thus it is of crucial importance to design efficient algorithms to perform the change of ordering. The main contributions of this paper are several efficient methods for the change of ordering which take advantage of the sparsity of multiplication matrices in the classical {\sf FGLM} algorithm. Combing all these methods, we propose a deterministic top-level algorithm that automatically detects which method to use depending on the input. As a by-product, we have a fast implementation that is able to handle ideals of degree over . Such an implementation outperforms the {\sf Magma} and {\sf Singular} ones, as shown by our experiments. First for the shape position case, two methods are designed based on the Wiedemann algorithm: the first is probabilistic and its complexity to complete the change of ordering is , where is the number of nonzero entries of a multiplication matrix; the other is deterministic and computes the LEX \grobner basis of via Chinese Remainder Theorem. Then for the general case, the designed method is characterized by the Berlekamp--Massey--Sakata algorithm from Coding Theory to handle the multi-dimensional linearly recurring relations. Complexity analyses of all proposed methods are also provided. Furthermore, for generic polynomial systems, we present an explicit formula for the estimation of the sparsity of one main multiplication matrix, and prove its construction is free. With the asymptotic analysis of such sparsity, we are able to show for generic systems the complexity above becomes
Design of termination criterion of BMS algorithm for lexicographical ordering
National audienceThe BMS algorithm from Coding Theory has good decoding efficiency and error-correcting capability, and current work usually focuses on the case of graded orderings. With the analysis on the essential characteristics of lexicographical and graded orderings, a termination criterion of BMS algorithm for the lexicographical ordering is designed by using the elimination property of Gröbner bases, which are closely related to the BMS algorithm, and an implementation-oriented description for the algorithm based on this criterion is also provided. Experimental results verify the effectiveness of the proposed termination criterion, and the new criterion matches the original theoretical one in the algorithm
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Polynomial system solving over finite fields is of particular interest because of its applications in Cryptography, Coding Theory, and other areas of information science and technologies. In this thesis we study several important theoretical and computational aspects for solving polynomial systems over finite fields, in particular on the two widely used tools Gröbner bases and triangular sets. We propose efficient algorithms for change of ordering of Gröbner bases of zero-dimensional ideals by using the sparsity of multiplication matrices and evaluate such sparsity for generic polynomial systems. Original algorithms are presented for decomposing polynomial sets into simple triangular sets over finite fields. We also define squarefree decomposition and factorization of polynomials over unmixed products of field extensions and propose algorithms for computing them. The effectiveness and efficiency of these algorithms have been verified by experiments with our implementations. Methods for polynomial system solving over finite fields are also applied to solve practical problems arising from Biology and Coding Theory.RĂ©solution de systĂšmes polynomiaux sur les corps finis est dâun intĂ©rĂȘt particulier en raison de ses applications en Cryptographie, ThĂ©orie du Codage, et dâautres domaines de la science de lâinformation. Dans cette thĂšse, nous Ă©tudions plusieurs aspects importants thĂ©oriques et informatiques pour rĂ©solution de systĂšmes polynomiaux sur les corps finis, en particulier sur les deux outils largement utiliss: bases de Gröbner et ensembles triangulaires. Nous proposons des algorithmes efficaces pour le changement de lâordre des bases de Gröbner dâidĂ©aux de dimension zĂ©ro en utilisant le faible densitĂ© des matrices de multiplication et dâĂ©valuer telle faible densitĂ© pour les systĂšmes de polynĂŽmes gĂ©nĂ©riques. Algorithmes originaux sont prĂ©sentĂ©s pour la dĂ©composition des ensembles de polynĂŽmes en ensembles triangulaires simples sur les corps finis. Nous dĂ©finissons Ă©galement dĂ©composition sans carrĂ© et factorisation des polynĂŽmes sur produits non mĂ©langĂ©s dâextensions des corps et proposons des lgorithmes pour les calculer. LâefficacitĂ© et lâefficience de ces algorithmes ont Ă©tĂ© vĂ©rifiĂ©es par des expĂ©riences avec nos implĂ©mentations. MĂ©thodes de rĂ©solution de systĂšmes polynomiaux sur les corps finis sont Ă©galement appliquĂ©es pour rĂ©soudre les problĂšmes pratiques posĂ©s par la Biologie et la ThĂ©orie du Codage
Solving Polynomial Systems over Finite Fields: Algorithms, Implementation and Applications
Polynomial system solving over finite fields is of particular interest because of its applications in Cryptography, Coding Theory, and other areas of information science and technologies. In this thesis we study several important theoretical and computational aspects for solving polynomial systems over finite fields, in particular on the two widely used tools Gröbner bases and triangular sets. We propose efficient algorithms for change of ordering of Gröbner bases of zero-dimensional ideals by using the sparsity of multiplication matrices and evaluate such sparsity for generic polynomial systems. Original algorithms are presented for decomposing polynomial sets into simple triangular sets over finite fields. We also define squarefree decomposition and factorization of polynomials over unmixed products of field extensions and propose algorithms for computing them. The effectiveness and efficiency of these algorithms have been verified by experiments with our implementations. Methods for polynomial system solving over finite fields are also applied to solve practical problems arising from Biology and Coding Theory.RĂ©solution de systĂšmes polynomiaux sur les corps finis est dâun intĂ©rĂȘt particulier en raison de ses applications en Cryptographie, ThĂ©orie du Codage, et dâautres domaines de la science de lâinformation. Dans cette thĂšse, nous Ă©tudions plusieurs aspects importants thĂ©oriques et informatiques pour rĂ©solution de systĂšmes polynomiaux sur les corps finis, en particulier sur les deux outils largement utiliss: bases de Gröbner et ensembles triangulaires. Nous proposons des algorithmes efficaces pour le changement de lâordre des bases de Gröbner dâidĂ©aux de dimension zĂ©ro en utilisant le faible densitĂ© des matrices de multiplication et dâĂ©valuer telle faible densitĂ© pour les systĂšmes de polynĂŽmes gĂ©nĂ©riques. Algorithmes originaux sont prĂ©sentĂ©s pour la dĂ©composition des ensembles de polynĂŽmes en ensembles triangulaires simples sur les corps finis. Nous dĂ©finissons Ă©galement dĂ©composition sans carrĂ© et factorisation des polynĂŽmes sur produits non mĂ©langĂ©s dâextensions des corps et proposons des lgorithmes pour les calculer. LâefficacitĂ© et lâefficience de ces algorithmes ont Ă©tĂ© vĂ©rifiĂ©es par des expĂ©riences avec nos implĂ©mentations. MĂ©thodes de rĂ©solution de systĂšmes polynomiaux sur les corps finis sont Ă©galement appliquĂ©es pour rĂ©soudre les problĂšmes pratiques posĂ©s par la Biologie et la ThĂ©orie du Codage
Fast algorithm for change of ordering of zero-dimensional Gröbner bases with sparse multiplication matrices
International audienceLet I in K[x1,...,xn] be a 0-dimensional ideal of degree D where K is a field. It is well-known that obtaining efficient algorithms for change of ordering of Gröbner bases of I is crucial in polynomial system solving. Through the algorithm FGLM, this task is classically tackled by linear algebra operations in K[x1,...,xn]/I. With recent progress on Gröbner bases computations, this step turns out to be the bottleneck of the whole solving process. Our contribution is an algorithm that takes advantage of the sparsity structure of multiplication matrices appearing during the change of ordering. This sparsity structure arises even when the input polynomial system defining I is dense. As a by-product, we obtain an implementation which is able to manipulate 0-dimensional ideals over a prime field of degree greater than 30000. It outperforms the Magma/Singular/FGb implementations of FGLM. First, we investigate the particular but important shape position case. The obtained algorithm performs the change of ordering within a complexity O(D(N1+nlog(D))), where N1 is the number of nonzero entries of a multiplication matrix. This almost matches the complexity of computing the minimal polynomial of one multiplication matrix. Then, we address the general case and give corresponding complexity results. Our algorithm is dynamic in the sense that it selects automatically which strategy to use depending on the input. Its key ingredients are the Wiedemann algorithm to handle 1-dimensional linear recurrence (for the shape position case), and the Berlekamp-Massey-Sakata algorithm from Coding Theory to handle multi-dimensional linearly recurring sequences in the general case
Decomposing polynomial sets into simple sets over finite fields: The positive-dimensional case
International audienceThis paper presents an algorithm for decomposing any positive-dimensional polynomial set into simple sets over an arbitrary finite field. The algorithm is based on some relationship established between simple sets and radical ideals, reducing the decomposition problem to the problem of computing the radicals of certain ideals. In addition to direct application of the algorithms of Matsumoto and Kemper, the algorithm of Fortuna and others is optimized and improved for the computation of radicals of special ideals. Preliminary experiments with an implementation of the algorithm in Maple and Singular are carried out to show the effectiveness and efficiency of the algorithm