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Algebraic theory of locally nilpotent derivations

Author: Gene Freudenburg
Publisher: [Place of publication not identified] : Springer, 2017.
Series: Encyclopaedia of mathematical sciences, v. 136.; Encyclopaedia of mathematical sciences., Invariant theory and algebraic transformation groups ;, 7.
Edition/Format:   eBook : Document : EnglishView all editions and formats
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Invariant Theory and Algebraic Transformation Groups VII.
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Genre/Form: Electronic books
Additional Physical Format: (OCoLC)988283787
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: Gene Freudenburg
ISBN: 9783662553503 3662553503
OCLC Number: 1003701846
Description: 1 online resource.
Contents: Acknowledgments; Introduction; Historical Overview; Notes on the Second Edition; Contents; 1 First Principles; 1.1 Preliminaries; 1.1.1 Rings and Modules; 1.1.2 Fields; 1.1.3 Localizations; 1.1.4 Degree Functions; 1.1.5 Graded Rings and Homogeneous Derivations; 1.1.6 Associated Graded Rings; 1.1.7 Locally Finite and Locally Nilpotent Derivations; 1.1.8 Degree Function Induced by a Derivation; 1.1.9 Exponential and Dixmier Maps; 1.1.10 Derivative of a Polynomial; 1.2 Basic Facts About Derivations; 1.2.1 Algebraic Operations; 1.2.2 Subalgebra Nil(D); 1.2.3 Kernels; 1.2.4 Localization 1.2.5 Integral Ideals1.2.6 Extension of Scalars; 1.2.7 Integral Extensions and Conductor Ideals; 1.3 Varieties and Group Actions; 1.4 First Principles for Locally Nilpotent Derivations; 1.5 Ga-Actions; 1.5.1 Correspondence with LNDs; 1.5.2 Orbits, Vector Fields and Fixed Points; 1.6 Degree Resolution and Canonical Factorization; 1.6.1 Degree Modules; 1.6.2 Degree Resolutions; 1.6.3 Equivariant Affine Modifications; 1.6.4 Canonical Factorizations; 2 Further Properties of LNDs; 2.1 Irreducible Derivations; 2.2 Minimal Local Slices; 2.3 Four Lemmas About UFDs; 2.4 Degree of a Derivation 2.5 Makar-Limanov Invariant2.6 Quasi-Extensions and Zn-Gradings; 2.7 G-Critical Elements; 2.8 AB and ABC Theorems; 2.9 Cables and Cable Algebras; 2.9.1 Associated Rooted Tree; 2.9.2 D-Cables; 2.9.3 Cable Algebras; 2.10 Exponential Automorphisms; 2.11 Transvectants and Wronskians; 2.11.1 Transvectants; 2.11.2 Wronskians; 2.12 Recognizing Polynomial Rings; 3 Polynomial Rings; 3.1 Variables, Automorphisms, and Gradings; 3.1.1 Linear Maps and Derivations; 3.1.2 Triangular and Tame Automorphisms; 3.2 Derivations of Polynomial Rings; 3.2.1 Definitions; 3.2.2 Partial Derivatives 3.2.3 Jacobian Derivations3.2.4 Homogenizing a Derivation; 3.2.5 Other Base Rings; 3.3 Locally Nilpotent Derivations of Polynomial Rings; 3.4 Slices in Polynomial Rings; 3.5 Triangular Derivations and Automorphisms; 3.6 Group Actions on An; 3.6.1 Terminology; 3.6.2 Translations; 3.6.3 Planar Actions; 3.6.4 Theorem of Deveney and Finston; 3.6.5 Proper and Locally Trivial Ga-Actions; 3.7 Ga-Actions Relative to Other Group Actions; 3.8 Some Important Early Examples; 3.8.1 Bass's Example ([12], 1984); 3.8.2 Popov's Examples ([344], 1987); 3.8.3 Smith's Example ([386], 1989) 3.8.4 Winkelmann's Example 1 ([421], 1990)3.8.5 Winkelmann's Example 2 ([421], 1990); 3.8.6 Example of Deveney and Finston ([104], 1995); 3.9 Homogeneous Dependence Problem; 3.9.1 Construction of Examples; 3.9.2 Derksen's Example; 3.9.3 De Bondt's Examples; 3.9.4 Rank-4 Example in Dimension 5; 4 Dimension Two; 4.1 Background; 4.2 Newton Polygons; 4.3 Polynomial Ring in Two Variables Over a Field; 4.3.1 Rentschler's Theorem: First Proof; 4.3.2 Rentschler's Theorem: Second Proof; 4.3.3 Proof of Jung's Theorem; 4.3.4 Proof of Structure Theorem
Series Title: Encyclopaedia of mathematical sciences, v. 136.; Encyclopaedia of mathematical sciences., Invariant theory and algebraic transformation groups ;, 7.
Responsibility: Gene Freudenburg.

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