Valuation Theory in Interaction PDF
Preview Valuation Theory in Interaction
EMS Series of Congress Reports EMS Congress Reports publishes volumes originating from conferences or seminars focusing on any field of pure or applied mathematics. The individual volumes include an introduction into their subject and review of the contributions in this context. Articles are required to undergo a refereeing process and are accepted only if they contain a survey or significant results not published elsewhere in the literature. Previously published: Trends in Representation Theory of Algebras and Related Topics, Andrzej Skowron´ski (ed.) K-Theory and Noncommutative Geometry, Guillermo Cortiñas et al. (eds.) Classification of Algebraic Varieties, Carel Faber, Gerard van der Geer and Eduard Looijenga (eds.) Surveys in Stochastic Processes, Jochen Blath, Peter Imkeller and Sylvie Rœlly (eds.) Representations of Algebras and Related Topics, Andrzej Skowron´ski and Kunio Yamagata (eds.) Contributions to Algebraic Geometry. Impanga Lecture Notes, Piotr Pragacz (ed.) Geometry and Arithmetic, Carel Faber, Gavril Farkas and Robin de Jong (eds.) Derived Categories in Algebraic Geometry. Toyko 2011, Yujiro Kawamata (ed.) Advances in Representation Theory of Algebras, David J. Benson, Henning Krause and Andrzej Skowron´ski (eds.) Valuation Theory in Interaction Antonio Campillo Franz-Viktor Kuhlmann Bernard Teissier Editors Editors: Prof. Antonio Campillo Prof. Franz-Viktor Kuhlmann Departamento de Álgebra, Geometría y Topología Department of Mathematics and Statistics Universidad de Valladolid University of Saskatchewan Paseo de Belén 7 106 Wiggins Road 47011 Valladolid Saskatoon, SK, S7N 5E6 Spain Canada Email: [email protected] Email: [email protected] Prof. Bernard Teissier Institut de Mathématiques de Jussieu – Paris Rive Gauche UMR 7586 du CNRS Bâtiment Sophie Germain, Case 7012 75205 Paris Cedex 13 France [email protected] 2010 Mathematics Subject Classification: primary: 03CXX, 12JXX, 12E30, 12F10, 13A18, 14H20, 14M25; secondary: 06FXX, 11SXX, 11U09, 12DXX, 12E05, 12F05, 12GXX, 12L12, 13D40, 13F30, 13H05, 13JXX, 13N15, 14BXX, 14C20, 14EXX, 14F10, 14J17, 14HXX, 14PXX, 16W60, 32P05, 32SXX, 37A05, 54F50. Key words: valuation, defect, Abhyankar valuation, divisorial valuation, completion, local uniformization, toric geometry, key polynomial, excellent ring, local ring, valuation centered at a local domain, valuative tree, dicritical divisor, Rees valuation, Izumi’s theorem, plane curve singularity, Newton tree, rational surface singularity, Whitney stratification, jet scheme, embedded Nash problem, higher local field, wild ramification, dynamical system, irreducible polynomial, additive polynomial, Hilbertian field, large field, Galois theory, C-minimality, cell decomposition, imaginary element, formally real field, R-place, Hardy field, exponential-logarithmic series field, asymptotic integration, Hahn field, truncation, quasi-valuation. ISBN 978-3-03719-149-1 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © European Mathematical Society 2014 Contact address: European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum SEW A27 CH-8092 Zürich Switzerland Phone: +41 (0)44 632 34 36 / Email: [email protected] / Homepage: www.ems-ph.org Cover graphic: María Alberich, Universitat Politècnica de Catalunya (Barcelona) Typeset by the editors using the authors’ TeX files Printing and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany ∞ Printed on acid free paper 9 8 7 6 5 4 3 2 1 The authors and editors dedicate this volume to the memory of Ram Abhyankar, who contributed so much to valuation theory and singularity theory, among other fields, and who was an active participant at both International Valuation Theory Conferences. He passed away in November of 2012. Preface TheSecondInternationalConferenceandWorkshoponValuationTheorywasheld inSegoviaandElEscorial, Spain, July18-29, 2011. Itcoveredseveraloftherecent developments in valuation theory and its applications. Valuation theory developed in the first half of the last century as a part of algebraic number theory, the theory oforderedfieldsandthetheoryoforderedabeliangroups. Then, throughthework of Krull, Zariski and Abhyankar, valuation theory found important applications in algebraic geometry. Later, as the expected applications in algebraic geometry were obtained by other methods, and algebraic number theory did not provide any striking new valuation theoretical questions, research in the topic became less intense. However, mathematicians from real algebra and from model theory kept a keen interest in valuation theory, and there was a remarkable development of the subject, albeit somewhat distant from the main stream of algebraic geometry. In recent years, valuation theory has found its way back into algebraic geometry and many important applications in various branches of mathematics have been discovered. It is fascinating to see how fundamental principles of valuation theory are found to play a role in various topics which seem to be only loosely related. Asimportantexamples,valuationtheoryhasbecomeextremelyusefulinthethe- ory of complex dynamical systems, and in the study of non-oscillating trajectories ofrealanalyticvectorfieldsinthreedimensions. AnaloguesoftheRiemann-Zariski valuationspaceshavebeenfoundtobetheproperframeworkforquestionsofinter- section theory in algebraic geometry and in the analysis of singularities of complex plurisubharmonic functions. Inadifferentdirection,therelationbetweenBerkovichgeometry,tropicalgeom- etry and valuation spaces, on the one hand, and the geometry of arc spaces and valuation spaces, on the other, have begun to deepen and clarify. Eversinceitsbeginnings,valuationtheoryandGaloistheoryhavegrownclosely together and influenced each other. Arguably, studying and understanding the extensions of valuations in algebraic field extensions is one of the most important questions in valuation theory, whereas using valuation theory is one of he most important tools in studying Galois extensions of fields, as well as constructing field extensions with given properties. The well established topic of the model theory of valued fields is also being transformed, in particular through the study of valued fields with functions and operators, and through the study of types over valued fields. Simultaneously, the work on the “classical” aspects of valuation theory has in- creased and substantial progress has been made toward the solution of the local uniformization problem in arbitrary characteristic, the study of ramification of valuations in arbitrary dimension, and in particular the nature and role of the mysterious “defect”. All of this corresponds to new insights and new approaches to old (and often longstanding)problems. Thisiswhyrightnowisaperfecttimetoholdconferences on the newly growing and flourishing subject of valuation theory. The organizers of the Second International Conference and Workshop on Val- uation Theory chose to privilege the topics listed above among a great diversity of topics where valuation theory plays a role, but some of those other topics were represented as well, and some contributed lectures were accepted. The audience includedgraduatestudentsandresearchersinthevariousdirectionsofthissubject. Theconferencebroughttogethertheexpertsofseveralbranchesofvaluationtheory and of related topics in order to strengthen the relations between these branches. Atthesametimeitintroducednon-expertstotheprinciplesandresultsofvaluation viii theory, in particular when they were coming from areas of mathematics in which valuationtheoryhasrecentlywitnessedimportantcontributions(e.g.,algebraicge- ometry,Galoistheory,asymptoticanalysis,dynamicalsystems)orinwhichtherole ofvaluationtheorystillhastobedetermined(e.g., C*-algebras). Onemaingoalof our conference was to point out and study the relations between different branches of valuation theory, and their applications to other branches of mathematics. An- other main goal was to state and discuss the many important open problems in valuationtheoryinordertoprovideanoptimalbasisforfutureresearchconducted by experts and students. The scientific committee of the conference consisted of Bernard Teissier (chair), Antonio Campillo, Zoe Chatzidakis, Dale Cutkosky, Lou van den Dries, Charles Favre, Ehud Hrushovski, Kiran Kedlaya and Francois Loeser. The members of the organizingcommitteewereFranz-ViktorKuhlmann(chair),F´elixDelgado,Antonio D´ıaz-Cano, Ignacio Luengo, Florian Pop and Michel Vaqui´e. The local organizing committeeincludedSantiagoEncinas(chair),MariaAlberich,IgnacioFarr´an,Car- los Galindo, Evelia Garc´ıa Barroso and Pedro Gonz´alez P´erez. We gratefully acknowledge the financial and administrative support of the fol- lowing institutions, which has made the conference and thus also this proceed- ings volume possible: Projet ANR “Espaces de Berkovich” (Agence Nationale de la Recherche, France), Compositio Mathematica Foundation, Science and Innova- tionMinistrySpain(GrantreferenceMTM2010-12174-E),Consolideri-mathGrant (SARE-C6-0372), UniversidaddeValladolid, UniversidadComplutensedeMadrid, Universitat Polit`ecnica de Catalunya, Junta de Castilla y Le´on. June 2014 Antonio Campillo DepartamentodeA´lgebrayGeometr´ıayTopolog´ıa,Universidad de Valladolid, P◦ Belen 7, SP-47011 Valladolid, Spain [email protected] Franz-Viktor Kuhlmann Department of Mathematics and Statistics, University of Saskatchewan, 106 Wiggins Road, Saskatoon, SK, S7N 5E6, Canada [email protected] Bernard Teissier Institut de Math´ematiques de Jussieu – Paris Rive Gauche, UMR 7586 du CNRS, Baˆtiment Sophie Germain, Case 7012, 75205 PARIS Cedex 13, France [email protected] Contents Preface ..................................................................... vii Introduction ................................................................. xi A study of irreducible polynomials over henselian valued fields via distinguished pairs K. Aghigh, A. Bishnoi, S. Kumar and S. K. Khanduja ........................ 1 On fields of totally S-adic numbers — With an appendix by Florian Pop Lior Bary-Soroker and Arno Fehm ........................................... 11 Infinite towers of Artin-Schreier defect extensions of rational function fields Anna Blaszczok .............................................................. 16 A refinement of Izumi’s Theorem S´ebastien Boucksom, Charles Favre and Mattias Jonsson ..................... 55 Multivariable Hodge theoretical invariants of germs of plane curves. II Pierrette Cassou-Nogu`es and Anatoly Libgober ............................... 82 Existence des diviseurs dicritiques, d’apr`es S.S. Abhyankar Vincent Cossart, Micka¨el Matusinski and Guillermo Moreno-Soc´ıas ......... 136 Invariants of the graded algebras associated to divisorial valuations dominating a rational surface singularity Vincent Cossart, Olivier Piltant and Ana J. Reguera ........................ 148 An introduction to C-minimal structures and their cell decomposition theorem Pablo Cubides Kovacsics .................................................... 167 Valuation semigroups of Noetherian local domains Steven Dale Cutkosky ....................................................... 208 Additive polynomials over perfect fields Salih Durhan ............................................................... 219 On R-places and related topics Danielle Gondard-Cozette ................................................... 226 Extending valuations to formal completions F. J. Herrera Govantes, M. A. Olalla Acosta, M. Spivakovsky and B. Teissier 252 Extending real valuations to skew polynomial rings A. Granja, M. C. Mart´ınez and C. Rodr´ıguez ............................... 266 Stratifications in valued fields Immanuel Halupczok ........................................................ 288 Imaginaries and definable types in algebraically closed valued fields Ehud Hrushovski ............................................................ 297 x Defects of algebraic function fields, completion defects and defect quotients Franz-Viktor Kuhlmann and Asim Naseem .................................. 320 Ongeneralizedseriesfieldsandexponential-logarithmicseriesfieldswithderivations Micka¨el Matusinski ......................................................... 350 Jet schemes of rational double point singularities Hussein Mourtada .......................................................... 373 Valuations centered at a two-dimensional regular local domain: infima and topolo- gies Josnei Novacoski ........................................................... 389 Reduction of local uniformization to the rank one case Josnei Novacoski and Mark Spivakovsky ..................................... 404 Little survey on large fields – old & new Florian Pop ................................................................ 432 Quasi-valuations – topology and the weak approximation theorem Shai Sarussi ................................................................ 464 Overweight deformations of affine toric varieties and local uniformization Bernard Teissier ............................................................ 474 Detecting valuations using small Galois groups Adam Topaz ................................................................ 566 Truncation in Hahn fields Lou van den Dries .......................................................... 579 The ergodicity of 1-Lipschitz transformations on 2-adic spheres Ekaterina Yurova ........................................................... 596 Ramification of higher local fields, approaches and questions Liang Xiao and Igor Zhukov ................................................ 600
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