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Quantitative Properties of Real Algebraic Singularities

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Quantitative Properties of Real Algebraic Singularities Ph. D thesis : Quantitative properties of real algebraic singularities Lionel F. Alberti Directeur de th`ese : Bernard Mourrain Codirecteur de th`ese : Georges Comte September 22, 2008 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 . 19 2.4.2 Counting the number of branches . 21 2.4.3 Conic structure and connection algorithm . 22 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 . 51 4.1.2 An extension of the moving the wall theorem . 57 4.2 Measuringtransversality. 67 4.3 Conicstructure ............................ 72 1 4.3.1 The classical setup of the conic structure theorem . 72 4.3.2 Astableversionofconicstructure . 75 4.4 Relating Euclidean distance and transversality . .... 79 4.5 A tentative triangulation procedure . 84 5 A sweeping method to triangulate surfaces in three-space 93 5.1 Notations ............................... 94 5.1.1 Thecontourcurve .. .. .. .. .. .. .. .. .. 95 5.1.2 Characteristicpointsonthesurface . 96 5.2 Topologyofthex-criticalsections. 97 5.3 Topologyofthecontourcurve. 98 5.4 The algorithm for singular algebraic surfaces . 101 5.4.1 Thealgorithm ........................ 101 5.4.2 Connectionalgorithm . 103 5.4.3 Triangulationalgorithm . 104 5.5 Whyweobtainatriangulation . 106 5.5.1 Computation of a Whitney stratification . 106 5.5.2 Connectionofthesections. 109 5.5.3 Theisotopy.......................... 111 5.6 Example................................114 5.7 Complexityandeffectiveness . 116 6 BoundonthelocalsumofBettinumbersofagerm 118 6.1 Presentationofthemainresult . 120 6.2 Counter-examples. 127 6.2.1 Necessityofpuredimensionality . 128 6.2.2 Necessity of controlling the tangent cone: 3d case . 129 6.2.3 Necessity of controlling the tangent cone: general case . 131 6.3 Thezero-dimensionalcase . 135 6.4 When the cut of the tangent cone is smooth and pure dimensional143 6.4.1 Sections of tangent cone and germ have same topology . 144 6.4.2 Proofoflemma6.4.4. 147 6.4.3 Proofofmaintheorem. 150 6.5 Applications.............................. 152 6.5.1 Computing the multiplicity . 153 6.5.2 Bound on the density and the Lipschitz-Killing invariants 161 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 way to go and many people have helped my 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 thanks 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 20th 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 donnera un r´esultat dans une centaine d’ann´ee ou bien qui requiet 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´e intrins`eque des ob- jets 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´epour 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´equi 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´eomorphique `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 don- ner une proc´edure g´en´erale de triangulation des objets semi-alg´ebriques par une technique de subdivision. Le chapitre 5 explore une autre approche que la sub- division pour trianguler la vari´et´e. Il s’appuie aussi sur des stratifications mais utilise une m´ethode de balayage `ala place d’une m´ethode de subdivision. Son domaine d’application est r´eduit aux vari´et´es alg´ebriques de R3 afin de pouvoir ´ecrire un algorithme complet de triangulation (sans aucune restriction sur la vari´et´e). Le chapitre final 6, donne une information topologique plus abstraite: une borne uniforme sur un nombre de composantes connexes pour un germe de vari´et´eanalytique. Cette borne peut `ason tour ˆetre utilis´ee pour obtenir une borne sur la densit´edu germe et de mani`ere plus g´en´erale sur ses invariants de Lipschitz-Killing. Ces quantit´ees sont reli´ees `ala complexit´eg´eometrique du germe. L’obtention d’une triangulation (possiblement approch´ee) d’un ensemble d´efini par des ´equations polynomiales est un probl`eme qui a re¸cu beaucoup d’attention depuis l’av`enement de l’`ere informatique. Le chapitre 3 pr´esente un algorithme qui fournit toujours une triangulation proche de l’hypersurface alg´ebrique. La pr´esentation des techniques existantes pour obtenir des triangulations approch´ees des vari´et´es alg´ebriques est donc faite en son d´ebut.
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