The Valuative Tree (lecture Notes In Mathematics)

this Volume Is Devoted To A Beautiful Object, Called The Valuative Tree And Designed As A Powerful Tool For The Study Of Singularities In Two Complex Dimensions. Its Intricate Yet Manageable Structure Can Be Analyzed By Both Algebraic And Geometric Means. Many Types Of Singularities, Including Those Of Curves, Ideals, And Plurisubharmonic Functions, Can Be Encoded In Terms Of Positive Measures On The Valuative Tree. The Construction Of These Measures Uses A Natural Tree Laplace Operator Of Independent Interest. Thisvolume is devoted to a beautiful object, called the valuative tree and designed as a powerful tool for the study of singularities in two complex dimensions. Its intricate yet manageable structure can be analyzed by both algebraic and geometric means. Many types of singularities, including those of curves, ideals, and plurisubharmonic functions, can be encoded in terms of positive measures on the valuative tree. The construction of these measures uses a natural tree Laplace operator of independent interest. TOC:Introduction.- 1. Generalities.- 2. MacLane's Method.- 3. Tree Structures.- 4. Valuations Through PuiseuxSeries.- 5. Topologies.- 6. The Universal Dual Graph.- 7. Tree Measures.- 8. Applications of the Tree Analysis.- Appendix.- References.- Index Front Matter....Pages I-XV Introduction....Pages 1-7 1. Generalities....Pages 9-24 2. MacLane’s Method....Pages 25-42 3. Tree Structures....Pages 43-80 4. Valuations Through Puiseux Series....Pages 81-96 5. Topologies....Pages 97-110 6. The Universal Dual Graph....Pages 111-150 7. Tree Measures....Pages 151-192 8. Applications of the Tree Analysis....Pages 193-209 Appendix....Pages 211-225 References....Pages 227-229 Index....Pages 231-234 Errata....Pages 241-241 Errata....Pages 241-243 Errata....Pages 243-243 Errata....Pages 243-243 Errata....Pages 243-244 "This volume is devoted to the valuative tree and designed as a powerful tool for the study of singularities in two complex dimensions."--BOOK JACKET