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Algebraic Approach to Differential Equations TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk Algebraic Approach to Differential Equations Bibliotheca Alexandrina, Alexandria, Egypt 12 – 24 November 2007 Edited by Lê Du˜ ng Tráng ICTP, Trieste, Italy World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. ALGEBRAIC APPROACH TO DIFFERENTIAL EQUATIONS Copyright © 2010 by The Abdus Salam International Centre for Theoretical Physics ISBN-13 978-981-4273-23-7 ISBN-10 981-4273-23-6 Printed in Singapore. v PREFACE Inoctober2007,the\AbdusSalam"InternationalCentreforTheoreticalPhysics (ICTP) organized a school in mathematics at the Biblioteca Alexandrina in Alexandria,Egypt.Fromthe3rdcenturyB.C.untilthe4thcenturyA.C.Alexan- dria was a centre for mathematics. Euclid, Diophante, Eratostene, Ptolemy, Hy- patiawereamongthosewhomadethefameofAlexandriaanditsantiquelibrary. The choice of the Biblioteca Alexandria was symbolic. With the reconstruction of the library it was natural that one also resumes the universal intellectual ex- change of the antique library. The will of the director of the Biblioteca, Ismael Seralgedin made that school possible. The topic of the school was \Algebraic approach of di(cid:11)erential equations". ThisspecialtopicwhichisattheconvergenceofAlgebra,GeometryandAnalysis waschosentogathermathematicians of di(cid:11)erentdisciplines inEgypt.Thistopic arises from the pioneer work of E. Kolchin, L. G(cid:23)arding, B. Malgrange and was formalizedbytheschoolofM.SatoinJapan.Thetechniquesusedareamongthe most recent andmodern techniquesof mathematics. Inthese lectures we give an elementary presentation of the subject. Applications are given and new areas of researcharealsohinted.Thisbookallowstounderstanddevelopmentsofthis.We hope that this book which gathers most of the lectures given in Alexandria will interestspecialistsandshowhowlineardi(cid:11)erentialsystemsarestudiednowadays. I especially thankthesecretaries Alessandra Bergamo and Mabilo Koutouof the mathematics section of ICTP and Anna Triolo of the publications section of ICTP for all the help they gave for the publication of this book. Lê Du~ng Tra(cid:19)ng Erratum Theschool on\Algebraic ApproachtoDi(cid:11)erentialEquations" was organized byLêDu~ngTra(cid:19)ngfromtheICTPandEgyptiancolleagues,ProfessorDarwish Mohamed Abdalla from Alexandria University, Professor Fahmy Mohamed fromAl-AzharUniversity,ProfessorYousifMohamedfromtheAmericanUni- versity in Cairo. Professor Ismail Idris from Ain Shams replaced Professor Fahmy who had to leave during the conference. Special thanks are going to Professor Mohamed Darwish for his dedication in organizing the school. vii ACKNOWLEDGMENTS The school was made possible with the help of Mr Mohamed El Faham, Deputy Director of the Bibliotheca Alexandrina, Ms Sahar Aly in charge of the international meetings, Ms Mariam Moussa, Ms Marva Elwakie, Ms Yasmin Maamoun,Ms OmneyaKamel,Ms AsmaaSolimanand Ms Samar Seoud, all from Bibliotheca Alexandrina. From the ICTP side, Ms Koutou Mabilo and Ms Alessandra Bergamo was in charge of the organisationof the school and Ms Anna Triolowas in charge of the publication of the proceedings. Lê Dung Tra(cid:19)ng Head of the mathematics section ICTP, Trieste, Italy ix CONTENTS Preface v Acknowledgments vii D-Modules in Dimension 1 1 L. Narva(cid:19)ez Macarro Modules Over the Weyl Algebra 52 F. J. Castro Jim(cid:19)enez Geometry of Characteristic Varieties 119 D. T. Lê and B. Teissier Singular Integrals and the Stationary Phase Methods 136 E. Delabaere Hypergeometric Functions and Hyperplane Arrangements 210 M. Jambu Bernstein-Sato Polynomials and Functional Equations 225 M. Granger Di(cid:11)erential Algebraic Groups 292 B. Malgrange 1 D-MODULES IN DIMENSION 1 L.NARVA(cid:19)EZMACARRO(cid:3) Departamento de A(cid:19)lgebra &Instituto de Matem(cid:19)aticas (IMUS) Universidad deSevilla, P.O.Box 1160, 41080 Sevilla,Spain (cid:3)E-mail: [email protected] Introduction These notes are issued from a course taught in the I.C.T.P. School on Algebraic Approach to Di(cid:11)erential Equations, held at Alexandria (Egypt) from November 12 through November 24, 2007. These notes are intended to guide the reader from the classical theory of linear di(cid:11)erential equations in one complex variable to the theory of D- modules. In the (cid:12)rst foursections we try to motivate the use of sheaves,in veryconcreteterms,tostateCauchytheoremandtoexpressthephenomena of analytic continuation of solutions. We also study multivalued solutions around singular points. In sections 5 and 6 we recall the classical result of Fuchs, the index theorem of Komatsu-Malgrange and Malgrange’s homo- logicalcharacterizationofregularity,whichisakeypointin understanding regularityinhigherdimension.Section7isextractedfromtheverynicepa- per2 ofJ.Brianc(cid:24)onandPh.Maisonobe.Itcontainsthedivisiontoolsonthe ring of (germs of) linear di(cid:11)erential operators in one variable. They allow ustoprove\almosteverything"on(complexanalytic)D-moduletheoryin dimension1fromthe classicalresults.Section 8triestomotivatethe point ofviewofhighersolutions,alandmarkinD-module theory.Sections9and 10 deal with holonomic D-modules and the general notion of regularity. Both sections are technically based on the division tools and so they are very speci(cid:12)c for the one dimensional case, but they give a good (cid:13)avor of the general theory. Section 11 is written in collaboration with F. Gudiel and it containsthe localversionof the Riemann-Hilbert correspondencein (cid:3)PartiallysupportedbyMTM2007{66929andFEDER. 2 dimension 1 stated in the paper13 with some complements. In section 12 we sketch the theory of D-modules on a Riemann surface. We would like to thank the organizers of the I.C.T.P. school, specially M. Darwish who took care of all practical (and very important) details, and Lê Du~ng Tra(cid:19)ng who conceived the school and took the heavy task of editing the lecture notes. 1. Cauchy Theorem Let U (cid:26) C be an open set. A complex linear di(cid:11)erential equation on U is given by dny dy a +(cid:1)(cid:1)(cid:1)+a +a y =g; (1) ndzn 1dz 0 where the a and g are holomorphic functions on U and y is an unknown i holomorphic function on U, which in case it exists is called a solution (on U) of the equation (1). If the function a does not vanishes identically, we n say that equation (1) has order n. Wheng=0in(1),wecallitanhomogeneouscomplexlineardi(cid:11)erential equation. In such a case, the solutions form a complex vector space, i.e. -) the product of any constant and any solution is again a solution. -) The sum of two solutions is again a solution. Remark 1.1. A very basic (and obvious) remark is that a complex linear di(cid:11)erentialequationonU as(1)determines,byrestriction,acomplexlinear di(cid:11)erential equation on any open subset V (cid:26)U and we may be interested insearchingitssolutions,notonlyonthewholeU,butonanyopensubset V (cid:26)U. If a (x)6=0 for all x2U, then equation (1) is equivalent (in the sense n that they have the same solutions) to dny dy dy +a0 +(cid:1)(cid:1)(cid:1)+a0 +a0y =g0; (2) dzn n(cid:0)1dz 1dz 0 where a0 = ai and g0 = g . i an an Equation (2) is still equivalent to a linear system of order 1 Y B 1 1 dY =AY +B; Y =0 ... 1; B =0 ... 1 (3) dz BY C BB C @ nA @ nA