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An Introduction to Constructive Algebraic Analysis and Its Applications Alban Quadrat An introduction to constructive algebraic analysis and its applications Alban Quadrat To cite this version: Alban Quadrat. An introduction to constructive algebraic analysis and its applications. [Research Report] RR-7354, INRIA. 2010, pp.237. inria-00506104 HAL Id: inria-00506104 https://hal.inria.fr/inria-00506104 Submitted on 27 Jul 2010 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE Une introduction à l’analyse algébrique constructive et à ses applications Alban Quadrat N° 7354 July 2010 Algorithms, Certification, and Cryptography apport de recherche ISSN 0249-6399 ISRN INRIA/RR--7354--FR+ENG Une introduction à l’analyse algébrique constructive et à ses applications Alban Quadrat ∗ Theme : Algorithms, Certification, and Cryptography Équipe-Projet AT-SOP Rapport de recherche n° 7354 — July 2010 — 237 pages Résumé : Ce texte est une extension des notes de cours que j’ai préparés pour les Journées Nationales de Calcul Formel qui ont eu lieu au CIRM, Luminy (France) du 3 au 7 Mai 2010. Le but principal de ce cours était d’introduire la communauté française du calcul formel à l’analyse algébrique constructive, et particulièrement à la théorie des D-modules algébriques, à ses applications à la théorie mathématique des systèmes et à ses implantations dans des logiciels de calcul formel tels que Maple ou GAP4. Parce que l’analyse algébrique est une théorie mathématique qui utilise différentes techniques venant de la théorie des modules, de l’algèbre homologique, de la théorie des faisceaux, de la géométrie algébrique et de l’analyse microlocale, il peut être difficile d’entrer dans ce domaine, nouveau et fascinant, des mathématiques. En effet, il existe peu de textes introductifs. Nous sommes rapidement conduits aux livres de Björk qui, à première vue, peuvent sembler difficiles aux membres de la communauté de calcul formel et aux mathématiciens appliqués. Je pense que le problème vient moins de la difficulté technique de la littérature existante que du manque d’introductions pédagogiques qui donnent une idée globale du domaine, montrent quels types de résultats et d’applications on peut attendre et qui développent les différents calculs à mener sur des exemples explicites. A leur humble niveau, ces notes de cours ont pour but de combler ce manque, tout du moins en ce qui concerne les idées de base de l’analyse algébrique. Puisque l’on ne peut enseigner bien que les choses que l’on a bien comprises, j’ai choisi de restreindre cette introduction à mes travaux sur les aspects constructifs de l’analyse algébrique et sur ses applications. Mots-clés : Analyse algébrique, systèmes linéaires d’équations aux dérivées partielles, systèmes linéaires fonctionnels, théorie des modules, algèbre homologique, algèbre constructive, calcul formel, théorie des systèmes, théorie du contrôle. ∗ INRIA Sophia Antipolis, 2004 Route des Lucioles BP 93, 06902 Sophia Antipolis Cedex, France, [email protected], http://www.sophia.inria.fr/members/Alban.Quadrat/index.html Centre de recherche INRIA Sophia Antipolis – Méditerranée 2004, route des Lucioles, BP 93, 06902 Sophia Antipolis Cedex Téléphone : +33 4 92 38 77 77 — Télécopie : +33 4 92 38 77 65 An introduction to constructive algebraic analysis and its applications Abstract: This text is an extension of lectures notes I prepared for les Journées Nationales de Calcul Formel held at the CIRM, Luminy (France) on May 3-7, 2010. The main purpose of these lectures was to introduce the French community of symbolic computation to the constructive approach to algebraic analysis and particularly to algebraic D-modules, its applications to math- ematical systems theory and its implementations in computer algebra systems such as Maple or GAP4. Since algebraic analysis is a mathematical theory which uses different techniques coming from module theory, homological algebra, sheaf theory, algebraic geometry, and microlocal anal- ysis, it can be difficult to enter this fascinating new field of mathematics. Indeed, there are very few introducing texts. We are quickly led to Björk’s books which, at first glance, may look diffi- cult for the members of the symbolic computation community and for applied mathematicians. I believe that the main issue is less the technical difficulty of the existing references than the lack of friendly introduction to the topic, which could have offered a general idea of it, shown which kind of results and applications we can expect and how to handle the different computations on explicit examples. To a very small extent, these lectures notes were planned to fill this gap, at least for the basic ideas of algebraic analysis. Since, we can only teach well what we have clearly understood, I have chosen to focus on my work on the constructive aspects of algebraic analysis and its applications. Key-words: Algebraic analysis, linear systems of partial differential equations, linear functional systems, module theory, homological algebra, constructive algebra, symbolic computation, sys- tems theory, control theory. An introduction to constructive algebraic analysis and its applications i Table des matières 1 Algebraic analysis approach to mathematical systems theory 7 1.1 Linear systems and finitely presented left D-modules ................ 8 1.2 Finite free resolutions and extension functor ..................... 13 1.3 Constructive study of module properties ....................... 24 1.4 Parametrizations of linear systems .......................... 36 1.5 Quillen-Suslin theorem and Stafford’s theorems ................... 44 1.6 Applications to multidimensional control theory ................... 59 2 Monge parametrizations and purity filtration 79 2.1 Baer’s extensions and Baer’s isomorphism ...................... 79 2.2 Monge parametrizations ................................ 88 2.3 Characteristic variety and dimensions . 100 2.4 Purity filtration of differential modules . 106 3 Factorization, reduction and decomposition problems 125 3.1 Homomorphisms between two finitely presented modules . 125 3.2 Computation of left D-homomorphisms . 130 3.3 Conservations laws of linear PD systems . 136 3.4 System equivalences ..................................140 3.5 Factorization problem .................................149 3.6 Reduction problem ...................................153 3.7 Decomposition of finitely presented left D-modules . 155 3.8 Decomposition problem ................................160 4 Serre’s reduction 167 4.1 Introduction .......................................167 4.2 Generalization of Serre’s theorem . 168 4.3 Equivalence to Serre’s reduction ............................177 5 Implementations 187 5.1 The OreModules package ..............................187 5.2 The Jacobson package ................................198 5.3 The QuillenSuslin package .............................201 5.4 The Stafford package ................................205 5.5 The PurityFiltration package . 207 5.6 The OreMorphisms package .............................215 5.7 The Serre package ..................................221 RR n° 7354 ii Alban Quadrat INRIA An introduction to constructive algebraic analysis and its applications 1 Introduction This text is an extension of lectures notes I prepared for les Journées Nationales de Calcul Formel held at the CIRM, Luminy (France) on May 3-7, 2010. The main purpose of these lectures was to introduce the French community of symbolic computation to the constructive approach to algebraic analysis and particularly to algebraic D-modules, its applications to mathematical systems theory and its implementations in computer algebra systems such as Maple or GAP4. Since algebraic analysis is a mathematical theory which uses different techniques coming from module theory, homological algebra, sheaf theory, algebraic geometry, and microlocal analysis, it can be difficult to enter this fascinating new field of mathematics. Indeed, there are very few introducing texts (to our knowledge, the best one is [66] with a few chapters of [13]). We are quickly led to Bjork’s first book [10] which, at first glance, may look difficult for the members of the symbolic computation community and for applied mathematicians. I believe that the main issue is less the technical difficulty than the lack of friendly introduction to the topic, which could have offered a general idea of it, shown which kind of results and applications we can expect and how to handle the different computations on explicit examples. Indeed, even if algebraic analysis aims at studying linear systems of algebraic or analytic partial differential equations (“the Courant-Hilbert ([23]) for the new generation” according to [46]), no examples illustrate the main results of the books [10, 11, 13, 45, 46, 66]. And when the term “applications” appears in the title of a book on algebraic analysis such as Bjork’s second book “Analytic D-modules and Applications” ([11]), the term “applications” has to be taken in the sense of applications to other pure fields of mathematics such as algebraic geometry, analytic
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