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Quasi-semi-stable representations
Fix K a p-adic field and denote by G_K its absolute Galois group. Let K_infty be the extension of K obtained by adding (p^n)-th roots of a fixed uniformizer, and G_\infty its absolute Galois group.Expand
Fast Multiplication for Skew Polynomials
TLDR
The algorithms improve the best known complexity for various arithmetics problems, and reaches the optimal asymptotic complexity bound for large degree. Expand
A new faster algorithm for factoring skew polynomials over finite fields
TLDR
An algorithm for the factorization of skew polynomials over finite fields is provided, which is faster than the previously known algorithm, which was due to Giesbrecht (1998). Expand
Computations with p-adic numbers
  • X. Caruso
  • Mathematics, Computer Science
  • ArXiv
  • 24 January 2017
TLDR
This document contains the notes of a lecture I gave at the "Journees Nationales du Calcul Formel" (JNCF) on January 2017 to discuss low-level algorithmics for p-adic numbers. Expand
Residues of skew rational functions and linearized Goppa codes
  • X. Caruso
  • Mathematics, Computer Science
  • ArXiv
  • 22 August 2019
TLDR
The main objective is to develop a theory of residues for skew rational functions (which are, by definition, the quotients of two skew polynomials) and prove a skew analogue of the residue formula and a skew analog of the classical formula of change of variables for residues. Expand
Linear Algebra over Z_p[[u]] and related rings
TLDR
The aim of this paper is to explain how to compute efficiently usual operations such as sum and intersection of sub-S-modules of S^d by computing an approximation of the result of these operations up to a quasi-isomorphism. Expand
Random matrices over a DVR and LU factorization
  • X. Caruso
  • Computer Science, Mathematics
  • J. Symb. Comput.
  • 1 November 2015
TLDR
It is proved that on average these valuations of the factors L and U in a discrete valuation ring are not too large and how one can apply this result to provide an efficient algorithm computing a basis of a coherent sheaf over A K 1 from the knowledge of its stalks is explained. Expand
Some Bounds for ramification of $p^n$-torsion semi-stable representations
Let p be an odd prime, K a finite extension of Q_p, G=Gal(\bar K/K) the Galois group and e=e(K/Q_p) the ramification index. Suppose T is a p^n torsion representation such that T is isomorphic to aExpand
Almost all non-archimedean Kakeya sets have measure zero
We study Kakeya sets over local non-archimedean fields with a probabilistic point of view: we define a probability measure on the set of Kakeya sets as above and prove that, according to thisExpand
Tracking p-adic precision
TLDR
A new method to propagate p-adic precision in computations, which also applies to other ultrametric fields, is presented and given a toy application to the stable computation of the SOMOS 4 sequence. Expand
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