18 research outputs found
Effective descent for differential operators
A theorem of N. Katz \cite{Ka} p.45, states that an irreducible differential
operator over a suitable differential field , which has an isotypical
decomposition over the algebraic closure of , is a tensor product
of an absolutely irreducible operator over and an
irreducible operator over having a finite differential Galois group.
Using the existence of the tensor decomposition , an algorithm is
given in \cite{C-W}, which computes an absolutely irreducible factor of
over a finite extension of . Here, an algorithmic approach to finding
and is given, based on the knowledge of . This involves a subtle descent
problem for differential operators which can be solved for explicit
differential fields which are -fields.Comment: 21 page
A Recursive Method for Determining the One-Dimensional Submodules of Laurent-Ore Modules
We present a method for determining the one-dimensional submodules of a
Laurent-Ore module. The method is based on a correspondence between
hyperexponential solutions of associated systems and one-dimensional
submodules. The hyperexponential solutions are computed recursively by solving
a sequence of first-order ordinary matrix equations. As the recursion proceeds,
the matrix equations will have constant coefficients with respect to the
operators that have been considered.Comment: To appear in the Proceedings of ISSAC 200
A Sage package for the symbolic-numeric factorization of linear differential operators
Software presentation accepted at ISSAC' 21 (Saint Petersburg, Russia, July 18-23, 2021)International audienceWe present a SageMath implementation of the symbolic-numeric algorithm introduced by van der Hoeven in 2007 for factoring linear differential operators whose coefficients are rational functions
Computing periods of rational integrals
A period of a rational integral is the result of integrating, with respect to
one or several variables, a rational function over a closed path. This work
focuses particularly on periods depending on a parameter: in this case the
period under consideration satisfies a linear differential equation, the
Picard-Fuchs equation. I give a reduction algorithm that extends the
Griffiths-Dwork reduction and apply it to the computation of Picard-Fuchs
equations. The resulting algorithm is elementary and has been successfully
applied to problems that were previously out of reach.Comment: To appear in Math. comp. Supplementary material at
http://pierre.lairez.fr/supp/periods