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Quantitative Properties of Real Algebraic Varieties Lionel Alberti Quantitative properties of real algebraic varieties Lionel Alberti To cite this version: Lionel Alberti. Quantitative properties of real algebraic varieties. Mathematics [math]. Université Nice Sophia Antipolis, 2008. English. tel-00449506 HAL Id: tel-00449506 https://tel.archives-ouvertes.fr/tel-00449506 Submitted on 21 Jan 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. THESE` pr´esent´ee DEVANT L’UNIVERSITE´ DE NICE SOPHIA-ANTIPOLIS pour obtenir le grade de DOCTEUR DE L’UNIVERSITE´ DE NICE SOPHIA-ANTIPOLIS Mention Math´ematiques par Lionel ALBERTI Universit´ede Nice Sophia-Antipolis, Institut National de Recherche en Informatique et Automatique Ecole´ Doctorale Sciences Fondamentales et Appliqu´ees U.F.R Math´ematiques TITRE DE LA THESE` : Propri´et´es des Singularit´es des Vari´et´es Alg´ebriques R´eelles Soutenue le 4 D´ecembre 2008 devant la Comission d’Examen COMPOSITION DU JURY : Mme Mari-Emi ALONSO Professor Examinatrice MSaugataBASU Professor Rapporteur M Georges COMTE Maitre de Conf´erences, HDR Co-directeur M Alexandru DIMCA Professeur Examinateur M Marc GIUSTI Professeur Examinateur M Bernard MOURRAIN Directeur de recherche, HDR Directeur M David TROTMAN Professeur Rapporteur Contents 1 Introduction 4 1.1 Fran¸cais................................ 4 1.2 English ................................ 7 2 Fast and certified topology computations for planar curves 11 2.1 Notationsanddefinitions . 13 2.2 Overview ............................... 14 2.3 Regulardomains ........................... 15 2.4 Simplysingulardomains. 19 2.4.1 TopologicalDegree . 20 2.4.2 Counting the number of branches . 22 2.4.3 Conic structure and connection algorithm . 23 2.5 Isolatingtheinterestingpoints . 24 2.5.1 Subdivisionmethod . 25 2.5.2 Rational univariate representation . 27 2.6 Examples ............................... 28 3 Triangulating smooth algebraic varieties 31 3.1 Backgroundonmeshing implicit surfaces. 31 3.2 Algebraicingredients. 34 3.2.1 Representationofpolynomials . 34 3.2.2 Univariatesolver . 35 3.3 Towardaguaranteedmethod . 36 3.3.1 Descriptionofthealgorithm. 36 3.3.2 Complexityanalysis . 40 3.3.3 Singularities.......................... 42 3.4 Experimentation ........................... 43 4 Whitney stratifications, Transversality, and Triangulations 45 4.1 Stratification and topological stability . ... 51 4.1.1 Notations and background on stratifications . 52 4.1.2 An extension of the moving the wall theorem . 58 4.1.3 A more practical form of the extended theorem . 66 4.2 Measuringtransversality. 76 1 4.3 Conicstructure ............................ 81 4.3.1 The classical setup of the conic structure theorem . 82 4.3.2 Astableversionofconicstructure . 84 4.4 Relating Euclidean distance and transversality . .... 88 4.5 A tentative triangulation procedure . 93 5 A sweeping method to triangulate surfaces in three-space 102 5.1 Notations ............................... 103 5.1.1 Thecontourcurve . 104 5.1.2 Characteristicpointsonthesurface . 105 5.2 Topologyofthex-criticalsections. 106 5.3 Topologyofthecontourcurve. 108 5.4 The algorithm for singular algebraic surfaces . 110 5.4.1 Thealgorithm ........................ 110 5.4.2 Connectionalgorithm . 111 5.4.3 Triangulationalgorithm . 113 5.5 Whyweobtainatriangulation . 114 5.5.1 Computation of a Whitney stratification . 116 5.5.2 Connectionofthesections. 118 5.5.3 Theisotopy.......................... 120 5.6 Example................................122 5.7 Complexityandeffectiveness . 125 6 BoundonthelocalsumofBettinumbersofagerm 127 6.1 Presentationofthemainresult . 129 6.2 Counter-examples. 136 6.2.1 Necessityofpuredimensionality . 137 6.2.2 Necessity of controlling the tangent cone: 3d case . 138 6.2.3 Necessity of controlling the tangent cone: general case . 140 6.3 Thezero-dimensionalcase . 144 6.4 When the cut of the tangent cone is smooth and pure dimensional152 6.4.1 Sections of tangent cone and germ have the same topology 153 6.4.2 Proofoflemma6.4.4. 156 6.4.3 Proofofmaintheorem. 159 6.5 Applications.............................. 161 6.5.1 Computing the multiplicity . 162 6.5.2 Bound on the density and the Lipschitz-Killing invariants 171 2 Remerciements : La pr´eparation d’une th`ese est un travail long et beau- coup de personnes m’ont aid´eau cours de celle-ci. Je voudrais remercier ma famille et mes amis, pour le soutien qu’ils m’ont donn´eaux moments difficiles, et aux moments moins difficiles aussi. Je voudrais aussi remercier les nombreuses personnes qui ont pris de leur temps pour r´epondre `ames questions et me faire partager leur connaissances, notam- ment G. Rond, E. Brugall´e, D. Trotman, G. Valette, et G. Lecerf. Finalement, mes plus grands remerciements vont `aceux qui m’ont encadr´edans la dur´ee : mes directeurs de th`ese Bernard Mourrain et Georges Comte, ainsi que le professeur Edward Bierstone qui m’a par deux fois accueilli `aToronto avec la plus grande gentillesse. Acknowledgements : Preparing a Ph. D thesis is a long process and many people helped me along it. I would like to thank my family and friends for the support they gave me in hard times, and in less hard times too. I would also like to thank the many persons who gave some of their time to answer my questions and share their knowledge with me, in particular G. Rond, E. Brugall´e, D. Trotman, G. Valette, et G. Lecerf. Finally, I am very grateful to those who have supervised my work on the long haul: my Ph. D advisers Bernard Mourrain and Georges Comte, as well as Prof. Edward Bierstone who most kindly received me twice in Toronto. 3 Chapter 1 Introduction 1.1 Fran¸cais L’arriv´ee r´ecente des ordinateurs dans tous les domaines des sciences a motiv´e beaucoup de recherches sur les aspects quantitatifs des objets math´ematiques. Cette voie fut `al’origine suivie par les constructivistes au d´ebut du 20i`eme si`ecle, mais ce n’est que r´ecemment que les th´eories quantitatives trouvent un large do- maine d’application. La transformation de th´eor`emes d’existence en algorithmes concrets a mˆeme cr´e´eune nouvelle activit´ede recherche connue sous le nom de math´ematiques exp´erimentales. En opposition `al’analyse et `ala g´eom´etrie diff´erentielle, le monde informatique ne consiste que de donn´ees discr`etes, ce qui fait de l’alg`ebre (discr`ete) et de la g´eom´etrie alg´ebrique des outils naturels pour les technologies de l’information. Les math´ematiques quantitatives ne se pr´eoccupent pas seulement de rendre des th´eor`emes effectifs, dans le sens de l’existence d’un algorithme, elles sont en fait principalement orient´ees vers la mise au point d’algorithmes rapides. Le temps et l’espace (m´emoire) sont importants car avoir un algorithme qui don- nera un r´esultat dans une centaine d’ann´ees ou bien qui requiert des millions de teraoctets de m´emoire ´equivaut en pratique `ane pas avoir d’algorithme du tout. Dans ce contexte, il devient important de comprendre la complexit´eintrins`eque des objets que l’on manipule afin d’´eviter de s’obstiner `amodifier un algorithme en esp´erant r´esoudre rapidement un probl`eme qui est en fait trop compliqu´e pour ˆetre trait´een un temps raisonnable. Cette th`ese s’int´eresse aux vari´et´es semi-alg´ebriques r´eelles ainsi qu’aux singu- larit´es qui apparaissent in´evitablement lorsqu’on les manipule. Bien ´evidemment, mˆeme en s’´etant restraint `ace domaine, le champ de recherche reste gigantesque. Les sujets abord´es dans ce travail ont donc ´et´echoisis selon les orientations de mes encadrants. Je pense qu’il est raisonnable de dire que cette th`ese explore les relations entre six principaux objets: la topologie et les triangulations, les tech- niques de subdivision, les stratifications, la complexit´e g´eom´etrique, la transver- salit´e, et la densit´e. Topologie et triangulation sont group´ees car dans un con- 4 texte informatique, l’encodage de la topologie par une triangulation est sans doute la seule repr´esentation universelle qui existe. La topologie/triangulation est le sujet des quatres premiers chapitres 2, 3, 4 et 5. Le chapitre 2 pr´esente un algorithme pour trianguler les courbes planes, et les outils utilis´es sont donc rel- ativement ´el´ementaires. La technique qui soutend l’approche du chapitre 2 est la subdivision du domaine. Le chapitre 3 pr´esente aussi une m´ethode de subdivi- sion pour calculer la topologie d’une hypersurface (discut´ee dans le cas des sur- faces). Cependant, la complexit´eg´eom´etrique des singularit´es pour des vari´et´es de dimension sup´erieure ou ´egale `a2 ne permet pas d’ais´ement les trianguler. Par cons´equent l’algorithme ne fournit une triangulation que lorsque la vari´et´e est lisse. D’autre part, la complexit´ede l’algorithme est estim´ee dans le cas lisse en fonction de la complexit´eg´eom´etrique de la surface. Lorsque la vari´et´eest singuli`ere l’algorithme permet d’isoler la partie singuli`ere aussi pr´ecis´ement que souhait´e, mais ne garanti pas que le complexe simplicial obtenu est en effet une triangulation hom´eomorphe `al’hypersurface. Afin de traiter ces probl`emes de singularit´e, les notions de stratification et de la transversalit´esont introduites dans le chapitre 4 pour analyser la topologie des ensembles semi-alg´ebriques dans Rn en g´en´eral. Stratification et transversalit´ejouent un rˆole important dans ce chapitre et ceux qui suivent (4, 5, 6). Dans le chapitre 4 elles sont utilis´ees pour donner une version ´etendue du th´eor`eme de Thom-Mather afin de donner une proc´edure g´en´erale de triangulation des objets semi-alg´ebriques par une technique de subdivision.
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