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Moulay A. Barkatou

Publications and source records attributed to Moulay A. Barkatou.

7 recordsLinked to original sources
arXiv2025

Gr{ö}bner bases over polytopal affinoid algebras

Polyhedral affinoid algebras have been introduced by Einsiedler, Kapranov and Lind to connect rigid analytic geometry (analytic geometry over non-archimedean fields) and tropical geometry. In this article, we present a theory of Gr{ö}bner bases for polytopal affinoid algebras that extends both Caruso et al.'s theory of Gr{ö}bner bases on Tate algebras and Pauer et al.'s theory of Gr{ö}bner bases on Laurent polynomials. We provide effective algorithms to compute Gr{ö}bner bases for both ideals of Laurent polynomials and ideals in polytopal affinoid algebras. Experiments with a Sagemath implementation are provided.

cs.SC↗
arXiv2019

Linear Differential Systems with Small Coefficients: Various Types of Solvability and their Verification

We study the problem of solvability of linear differential systems with small coefficients in the Liouvillian sense (or, by generalized quadratures). For a general system, this problem is equivalent to that of solvability of the Lie algebra of the differential Galois group of the system. However, dependence of this Lie algebra on the system coefficients remains unknown. We show that for the particular class of systems with non-resonant irregular singular points that have sufficiently small coefficient matrices, the problem is reduced to that of solvability of the explicit Lie algebra generated by the coefficient matrices. This extends the corresponding Ilyashenko-Khovanskii theorem obtained for linear differential systems with Fuchsian singular points. We also give some examples illustrating the practical verification of the presented criteria of solvability by using general procedures implemented in Maple.

math.CA↗
arXiv2018

Desingularization of First Order Linear Difference Systems with Rational Function Coefficients

It is well known that for a first order system of linear difference equations with rational function coefficients, a solution that is holomorphic in some left half plane can be analytically continued to a meromorphic solution in the whole complex plane. The poles stem from the singularities of the rational function coefficients of the system. Just as for differential equations, not all of these singularities necessarily lead to poles in solutions, as they might be what is called removable. In our work, we show how to detect and remove these singularities and further study the connection between poles of solutions and removable singularities. We describe two algorithms to (partially) desingularize a given difference system and present a characterization of removable singularities in terms of shifts of the original system.

cs.SC↗
arXiv2016

Formal Solutions of Completely Integrable Pfaffian Systems With Normal Crossings

In this paper, we present an algorithm for computing a fundamental matrix of formal solutions of completely integrable Pfaffian systems with normal crossings in several variables. This algorithm is a generalization of a method developed for the bivariate case based on a combination of several reduction techniques and is implemented in the computer algebra system Maple.

cs.SC↗