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Factoring Partial Differential Systems in Posi tive Characteristic

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27 Pages
English

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Niveau: Supérieur, Doctorat, Bac+8
Factoring Partial Differential Systems in Posi- tive Characteristic M. A. Barkatou, T. Cluzeau, J.-A. Weil with an appendix by M. van der Put: Classification of Partial Differential Modules in Positive Characteristic Abstract. An algorithm for factoring differential systems in characteristic p has been given by Cluzeau in [Cl03]. It is based on both the reduction of a matrix called p-curvature and eigenring techniques. In this paper, we gener- alize this algorithm to factor partial differential systems in characteristic p. We show that this factorization problem reduces effectively to the problem of simultaneous reduction of commuting matrices. In the appendix, van der Put shows how to extend his classification of differ- ential modules, used in the work of Cluzeau, to partial differential systems in positive characteristic. Mathematics Subject Classification (2000). 68W30; 16S32; 15A21; 16S50; 35G05. Keywords. Computer Algebra, Linear Differential Equations, Partial Differ- ential Equations, D-Finite Systems, Modular Algorithms, p-Curvature, Fac- torization, Simultaneous Reduction of Commuting Matrices. Introduction The problem of factoring D-finite partial differential systems in characteristic zero has been recently studied by Li, Schwarz and Tsarev in [LST02, LST03] (see also [Wu05]). In these articles, the authors show how to adapt Beke's algorithm (which factors ordinary differential systems, see [CH04] or [PS03, 4.2.1] and references therein) to the partial differential case.

  • positive characteristic

  • partial differential

  • partial differ- ential system

  • zero-dimensional poly- nomial

  • ential equations

  • differential system

  • system over


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Factoring Partial Differential Systems in tive Characteristic
M. A. Barkatou, T. Cluzeau, J.-A. Weil
with an appendix by M. van der Put: Classification of Partial Differential Modules in Positive Characteristic
Posi-
Abstract.An algorithm for factoring differential systems in characteristicp has been given by Cluzeau in [Cl03]. It is based on both the reduction of a matrix calledpeigenring techniques. In this paper, we gener--curvature and alize this algorithm to factor partial differential systems in characteristicp. We show that this factorization problem reduces effectively to the problem of simultaneous reduction of commuting matrices. In the appendix, van der Put shows how to extend his classification of differ-ential modules, used in the work of Cluzeau, to partial differential systems in positive characteristic.
Mathematics Subject Classification (2000).68W30; 16S32; 15A21; 16S50; 35G05. Keywords.Computer Algebra, Linear Differential Equations, Partial Differ-ential Equations, D-Finite Systems, Modular Algorithms,p-Curvature, Fac-torization, Simultaneous Reduction of Commuting Matrices.
Introduction
The problem of factoringD-finite partial differential systems in characteristic zero hasbeenrecentlystudiedbyLi,SchwarzandTsare¨vin[LST02,LST03](seealso [Wu05]). In these articles, the authors show how to adapt Beke’s algorithm (which factors ordinary differential systems, see [CH04] or [PS03, 4.2.1] and references therein) to the partial differential case. The topic of the present paper is an al-gorithm that factorsD-finite partial differential systems in characteristicp. Aside from its theoretical value, the interest of such an algorithm is its potential use as a first step in the construction of a modular factorization algorithm; in addition, it provides useful modular filters,e.g., for detecting the irreducibility of partial differential systems.
´ T.Cluzeau initiated this work while being a member of Laboratoirestix, Ecole polytechnique, 91128 Palaiseau Cedex, France.
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M. A. Barkatou, T. Cluzeau, J.-A. Weil
Concerning the ordinary differential case in characteristicp, factorization algorithms have been given by van der Put in [Pu95, Pu97] (see also [PS03, Ch.13]), Giesbrecht and Zhang in [GZ03] and Cluzeau in [Cl03, Cl04]. In this paper, we study the generalization of the one given in [Cl03]. Cluzeau’s method combines the use of van der Put’s classification of differential modules in characteristicpbased on thep-curvature (see [Pu95] or [PS03, Ch.13]) and the approach of the eigenring factorization method (see [Si96, Ba01, PS03]) as set by Barkatou in [Ba01]. In the partial differential case, we also have notions ofp-curvatures and eigen-rings at our disposal, but van der Put’s initial classification of differential modules in characteristicpcannot be applied directly, so we propose an alternative algorith-mic approach. To develop a factorization algorithm (and a partial generalization of van der Put’s classification) ofD-finite partial differential systems, we rebuild the elementary parts from [Cl03, Cl04] (where most proofs are algorithmic and independent of the classification) and generalize them to the partial differential context. In the appendix, van der Put develops a classification of “partial” differential mod-ules in positive characteristic which sheds light on our developments, and comes as a good complement to the algorithmic material elaborated in this paper. We follow the approach of [Cl03], that is, we first compute a maximal decom-position of our system before reducing the indecomposable blocks. The decompo-sition phase is separated into two distinct parts: we first use thep-curvature to compute asimultaneous decomposition(using a kind of ”isotypical decomposition” method), and then, we propose several methods to refine this decomposition into a maximal one. The generalization to the partial differential case amounts to applying si-multaneously the ordinary differential techniques to several differential systems. Consequently, since in the ordinary differential case we are almost always reduced to performing linear algebra on thep-curvature matrix, our generalization of the algorithm of [Cl03] relies on a way to reduce simultaneously commuting matrices (thep-curvatures). A solution to the latter problem has been sketched in [Cl04]; similar ideas can be found in papers dealing with numerical solutions of zero-dimensional poly-nomial systems such as [CGT97 ]. The essential results are recalled (and proved) here for self-containedness.
The paper is organized as follows. In the first part, we recall some defini-tions about (partial) differential systems and their factorizations. We then show how to generalize to the partial differential case some useful results concerning p-curvatures, factorizations and rational solutions of the system: we generalize the proofs given in [Cl03, Cl04]. After a section on simultaneous reduction of com-muting matrices, the fourth part contains the factorization algorithms. Finally, in Section 5, we show how the algorithm in [Cl03] can be directly generalized (with fewer efforts than for the partial differential case) to other situations: the case of “local” differential systems and that of difference systems.