Random matrix over a DVR and LU factorization

Let R be a discrete valuation ring (DVR) and K be its fraction field. If M is a matrix over R admitting a LU decomposition, it could happen that the entries of the factors L and U do not lie in R, but just in K. Having a good control on the valuations of these entries is very important for algorithmic applications. In the paper, we prove that in average these valuations are not too large and explain how one can apply this result to provide an efficient algorithm computing a basis of a coherent sheaf over A^1 from the knowledge of its stalks.Comment: 23 page

Resultants and subresultants of p-adic polynomials

We address the problem of the stability of the computations of resultants and subresultants of polynomials defined over complete discrete valuation rings (e.g. Zp or k[[t]] where k is a field). We prove that Euclide-like algorithms are highly unstable on average and we explain, in many cases, how one can stabilize them without sacrifying the complexity. On the way, we completely determine the distribution of the valuation of the principal subresultants of two random monic p-adic polynomials having the same degree

On Kant's first insight into the problem of space dimensionality and its physical foundations

In this article it is shown that a careful analysis of Kant's "Thoughts on the True Estimation of Living Forces" leads to a conclusion that does not match the usually accepted interpretation of Kant's reasoning in 1747, according to which the Young Kant supposedly establishes a relationship between the tridimensionality of space and Newton's law of universal gravitation. Indeed, it is argued that this text does not yield a satisfactory explanation of space dimensionality, actually restricting itself to justify the tridimensionality of extension.Comment: 14 page

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

Combinatoire du point de croix

This paper deals with a popular form of counted-thread embroidery: the cross-stitch. More precisely, we are interested in the minimal length of embroidery thread one need to fill a given picture. We give a complete answer to this problem for 4-connexe pictures (the notion is defined in the paper). In a last part, we study several other examples.Comment: 20 page

Tracking p-adic precision

We present a new method to propagate $p$-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

F_p-représentations semi-stables

37 pagesInternational audienceTorsion semi-stable representations can be constructed and studied using Breuil modules. In this paper, we define the notion of pylonet and prove that some categories of Breuil modules naturally define pylonets. As a consequence, we are able to define full subcategories of Breuil's categories with very nice properties (in particular, they are abelian). In a second part of this work, we try to make very explicit some abstract constructions coming from the general theory of pylonets (developped earlier in the paper). These explicitations should be very useful to make computations with torsion semi-stable Galois representations

Lettre à une amie fidèle

International audienceLe vocabulaire mathématique est certainement très technique, mais il est aussi souvent très imagé. Voici une lettre, sans prétention, que je me suis amusé à écrire pour illustrer ce propos. Pour les non-initiés, les explications de texte sont données en note de bas de page