29 research outputs found
Matrix-F5 algorithms over finite-precision complete discrete valuation fields
Let be a sequence
of homogeneous polynomials with -adic coefficients. Such system may happen,
for example, in arithmetic geometry. Yet, since is not an
effective field, classical algorithm does not apply.We provide a definition for
an approximate Gr{\"o}bner basis with respect to a monomial order We
design a strategy to compute such a basis, when precision is enough and under
the assumption that the input sequence is regular and the ideals are weakly--ideals. The conjecture of Moreno-Socias
states that for the grevlex ordering, such sequences are generic.Two variants
of that strategy are available, depending on whether one lean more on precision
or time-complexity. For the analysis of these algorithms, we study the loss of
precision of the Gauss row-echelon algorithm, and apply it to an adapted
Matrix-F5 algorithm. Numerical examples are provided.Moreover, the fact that
under such hypotheses, Gr{\"o}bner bases can be computed stably has many
applications. Firstly, the mapping sending to the reduced
Gr{\"o}bner basis of the ideal they span is differentiable, and its
differential can be given explicitly. Secondly, these hypotheses allows to
perform lifting on the Grobner bases, from to
or Finally, asking for the same
hypotheses on the highest-degree homogeneous components of the entry
polynomials allows to extend our strategy to the affine case
Matrix-F5 algorithms over finite-precision complete discrete valuation fields
International audienceLet be a sequence of homogeneous polynomials with -adic coefficients. Such system may happen, for example, in arithmetic geometry. Yet, since is not an effective field, classical algorithm does not apply.We provide a definition for an approximate Gröbner basis with respect to a monomial order We design a strategy to compute such a basis, when precision is enough and under the assumption that the input sequence is regular and the ideals are weakly--ideals. The conjecture of Moreno-Socias states that for the grevlex ordering, such sequences are generic.Two variants of that strategy are available, depending on whether one lean more on precision or time-complexity. For the analysis of these algorithms, we study the loss of precision of the Gauss row-echelon algorithm, and apply it to an adapted Matrix-F5 algorithm. Numerical examples are provided.Moreover, the fact that under such hypotheses, Gröbner bases can be computed stably has many applications. Firstly, the mapping sending to the reduced Gröbner basis of the ideal they span is differentiable, and its differential can be given explicitly. Secondly, these hypotheses allows to perform lifting on the Grobner bases, from to or Finally, asking for the same hypotheses on the highest-degree homogeneous components of the entry polynomials allows to extend our strategy to the affine case
Discriminants of Chebyshev Radical Extensions
Let t be any integer and fix an odd prime ell. Let Phi(x) = T_ell^n(x)-t
denote the n-fold composition of the Chebyshev polynomial of degree ell shifted
by t. If this polynomial is irreducible, let K = bbq(theta), where theta is a
root of Phi. A theorem of Dedekind's gives a condition on t for which K is
monogenic. For other values of t, we apply the Montes algorithm to obtain a
formula for the discriminant of K and to compute basis elements for the ring of
integers O_K.Comment: This update contains proofs for the conjectures appearing in a
earlier version of this paper. This article draws heavily from
arXiv:0906.262
Faster Sparse Matrix Inversion and Rank Computation in Finite Fields
We improve the current best running time value to invert sparse matrices over
finite fields, lowering it to an expected time for the
current values of fast rectangular matrix multiplication. We achieve the same
running time for the computation of the rank and nullspace of a sparse matrix
over a finite field. This improvement relies on two key techniques. First, we
adopt the decomposition of an arbitrary matrix into block Krylov and Hankel
matrices from Eberly et al. (ISSAC 2007). Second, we show how to recover the
explicit inverse of a block Hankel matrix using low displacement rank
techniques for structured matrices and fast rectangular matrix multiplication
algorithms. We generalize our inversion method to block structured matrices
with other displacement operators and strengthen the best known upper bounds
for explicit inversion of block Toeplitz-like and block Hankel-like matrices,
as well as for explicit inversion of block Vandermonde-like matrices with
structured blocks. As a further application, we improve the complexity of
several algorithms in topological data analysis and in finite group theory
Sur l'algorithme de décodage en liste de Guruswami-Sudan sur les anneaux finis
This thesis studies the algorithmic techniques of list decoding, first proposed by Guruswami and Sudan in 1998, in the context of Reed-Solomon codes over finite rings. Two approaches are considered. First we adapt the Guruswami-Sudan (GS) list decoding algorithm to generalized Reed-Solomon (GRS) codes over finite rings with identity. We study in details the complexities of the algorithms for GRS codes over Galois rings and truncated power series rings. Then we explore more deeply a lifting technique for list decoding. We show that the latter technique is able to correct more error patterns than the original GS list decoding algorithm. We apply the technique to GRS code over Galois rings and truncated power series rings and show that the algorithms coming from this technique have a lower complexity than the original GS algorithm. We show that it can be easily adapted for interleaved Reed-Solomon codes. Finally we present the complete implementation in C and C++ of the list decoding algorithms studied in this thesis. All the needed subroutines, such as univariate polynomial root finding algorithms, finite fields and rings arithmetic, are also presented. Independently, this manuscript contains other work produced during the thesis. We study quasi cyclic codes in details and show that they are in one-to-one correspondence with left principal ideal of a certain matrix ring. Then we adapt the GS framework for ideal based codes to number fields codes and provide a list decoding algorithm for the latter.Cette thèse porte sur l'algorithmique des techniques de décodage en liste, initiée par Guruswami et Sudan en 1998, dans le contexte des codes de Reed-Solomon sur les anneaux finis. Deux approches sont considérées. Dans un premier temps, nous adaptons l'algorithme de décodage en liste de Guruswami-Sudan aux codes de Reed-Solomon généralisés sur les anneaux finis. Nous étudions en détails les complexités de l'algorithme pour les anneaux de Galois et les anneaux de séries tronquées. Dans un deuxième temps nous approfondissons l'étude d'une technique de remontée pour le décodage en liste. Nous montrons que cette derni're permet de corriger davantage de motifs d'erreurs que la technique de Guruswami-Sudan originale. Nous appliquons ensuite cette même technique aux codes de Reed-Solomon généralisés sur les anneaux de Galois et les anneaux de séries tronquées et obtenons de meilleures bornes de complexités. Enfin nous présentons l'implantation des algorithmes en C et C++ des algorithmes de décodage en liste étudiés au cours de cette thèse. Tous les sous-algorithmes nécessaires au décodage en liste, comme la recherche de racines pour les polynômes univariés, l'arithmétique des corps et anneaux finis sont aussi présentés. Indépendamment, ce manuscrit contient d'autres travaux sur les codes quasi-cycliques. Nous prouvons qu'ils sont en correspondance biunivoque avec les idéaux à gauche d'un certain anneaux de matrices. Enfin nous adaptons le cadre proposé par Guruswami et Sudan pour les codes à base d'ideaux aux codes construits à l'aide des corps de nombres. Nous fournissons un algorithme de décodage en liste dans ce contexte
Arithmetical problems in number fields, abelian varieties and modular forms
La teoria de nombres, una à rea de la matemà tica fascinant i de les més antigues, ha experimentat un progrés espectacular durant els darrers anys. El desenvolupament d'una base teòrica profunda i la implementació d'algoritmes han conduït a noves interrelacions matemà tiques interessants que han fet palesos teoremes importants en aquesta à rea. Aquest informe resumeix les contribucions a la teoria de nombres dutes a terme per les persones del Seminari de Teoria de Nombres (UB-UAB-UPC) de Barcelona. Els seus resultats són citats en connexió amb l'estat actual d'alguns problemes aritmètics, de manera que aquesta monografia cerca proporcionar al públic lector una ullada sobre algunes lÃnies especÃfiques de la recerca matemà tica actual.Number theory, a fascinating area in mathematics and one of the oldest, has experienced spectacular progress in recent years. The development of a deep theoretical background and the implementation of algorithms have led to new and interesting interrelations with mathematics in general which have paved the way for the emergence of major theorems in the area. This report summarizes the contribution to number theory made by the members of the Seminari de Teoria de Nombres (UB-UAB-UPC) in Barcelona. These results are presented in connection with the state of certain arithmetical problems, and so this monograph seeks to provide readers with a glimpse of some specific lines of current mathematical research