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
A Tropical F5 algorithm
Let K be a field equipped with a valuation. Tropical varieties over K can be
defined with a theory of Gr{\"o}bner bases taking into account the valuation of
K. While generalizing the classical theory of Gr{\"o}bner bases, it is not
clear how modern algorithms for computing Gr{\"o}bner bases can be adapted to
the tropical case. Among them, one of the most efficient is the celebrated F5
Algorithm of Faug{\`e}re. In this article, we prove that, for homogeneous
ideals, it can be adapted to the tropical case. We prove termination and
correctness. Because of the use of the valuation, the theory of tropical
Gr{\"o}b-ner bases is promising for stable computations over polynomial rings
over a p-adic field. We provide numerical examples to illustrate
time-complexity and p-adic stability of this tropical F5 algorithm
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
Tracking p-adic precision
We present a new method to propagate -adic precision in computations,
which also applies to other ultrametric fields. We illustrate it with many
examples and give a toy application to the stable computation of the SOMOS 4
sequence
p-Adic Stability In Linear Algebra
Using the differential precision methods developed previously by the same
authors, we study the p-adic stability of standard operations on matrices and
vector spaces. We demonstrate that lattice-based methods surpass naive methods
in many applications, such as matrix multiplication and sums and intersections
of subspaces. We also analyze determinants , characteristic polynomials and LU
factorization using these differential methods. We supplement our observations
with numerical experiments.Comment: ISSAC 2015, Jul 2015, Bath, United Kingdom. 201
Précision p-adique
P-Adic numbers are a field in arithmetic analoguous to the real numbers. The advent during the last few decades of arithmetic geometry has yielded many algorithms using those numbers. Such numbers can only by handled with finite precision. We design a method, that we call differential precision, to study the behaviour of the precision in a p-adic context. It reduces the study to a first-order problem. We also study the question of which Gröbner bases can be computed over a p-adic number field.Les nombres p-adiques sont un analogue des nombres réels plus proche de l’arithmétique. L’avènement ces dernières décennies de la géométrie arithmétique a engendré la création de nombreux algorithmes utilisant ces nombres. Ces derniers ne peuvent être de manière générale manipulés qu’à précision finie. Nous proposons une méthode, dite de précision différentielle, pour étudier ces problèmes de précision. Elle permet de se ramener à un problème au premier ordre. Nous nous intéressons aussi à la question de savoir quelles bases de Gröbner peuvent être calculées sur les p-adiques
On the p-adic stability of the FGLM algorithm
Nowadays, many strategies to solve polynomial systems use the computation of a Gröbner basis for the graded reverse lexicographical ordering, followed by a change of ordering algorithm to obtain a Gröbner basis for the lexicographical ordering. The change of ordering algorithm is crucial for these strategies. We study the p-adic stability of the main change of ordering algorithm, FGLM. We show that FGLM is stable and give explicit upper bound on the loss of precision occuring in its execution. The variant of FGLM designed to pass from the grevlex ordering to a Gröbner basis in shape position is also stable. Our study relies on the application of Smith Normal Form computations for linear algebra
p-adic precision
Les nombres p-adiques sont un analogue des nombres réels plus proche de l’arithmétique. L’avènement ces dernières décennies de la géométrie arithmétique a engendré la création de nombreux algorithmes utilisant ces nombres. Ces derniers ne peuvent être de manière générale manipulés qu’à précision finie. Nous proposons une méthode, dite de précision différentielle, pour étudier ces problèmes de précision. Elle permet de se ramener à un problème au premier ordre. Nous nous intéressons aussi à la question de savoir quelles bases de Gröbner peuvent être calculées sur les p-adiques.P-Adic numbers are a field in arithmetic analoguous to the real numbers. The advent during the last few decades of arithmetic geometry has yielded many algorithms using those numbers. Such numbers can only by handled with finite precision. We design a method, that we call differential precision, to study the behaviour of the precision in a p-adic context. It reduces the study to a first-order problem. We also study the question of which Gröbner bases can be computed over a p-adic number field
p-adic precision
Les nombres p-adiques sont un analogue des nombres réels plus proche de l’arithmétique. L’avènement ces dernières décennies de la géométrie arithmétique a engendré la création de nombreux algorithmes utilisant ces nombres. Ces derniers ne peuvent être de manière générale manipulés qu’à précision finie. Nous proposons une méthode, dite de précision différentielle, pour étudier ces problèmes de précision. Elle permet de se ramener à un problème au premier ordre. Nous nous intéressons aussi à la question de savoir quelles bases de Gröbner peuvent être calculées sur les p-adiques.P-Adic numbers are a field in arithmetic analoguous to the real numbers. The advent during the last few decades of arithmetic geometry has yielded many algorithms using those numbers. Such numbers can only by handled with finite precision. We design a method, that we call differential precision, to study the behaviour of the precision in a p-adic context. It reduces the study to a first-order problem. We also study the question of which Gröbner bases can be computed over a p-adic number field