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Cover Art
PRINTED BOOKS
Author Freudenburg, Gene.

Title Algebraic theory of locally nilpotent derivations / Gene Freudenburg.

Published Berlin : Springer-Verlag, [2006]
©2006

Copies

Location Call No. Status
 UniM ERC  512.56 FREU    AVAILABLE
Physical description xi, 261 pages : illustrations ; 24 cm.
Series Encyclopaedia of mathematical sciences ; v. 136.
Invariant theory and algebraic transformation groups ; 7.
Encyclopaedia of mathematical sciences ; v. 136.
Encyclopaedia of mathematical sciences. Invariant theory and algebraic transformation groups ; 7.
Bibliography Includes bibliographical references and index.
Contents 1 First Principles 9 -- 1.1 Basic Definitions for Derivations 9 -- 1.2 Basic Facts about Derivations 15 -- 1.3 Group Actions 19 -- 1.4 First Principles for Locally Nilpotent Derivations 22 -- 1.5 G[subscript a]-Actions 31 -- 2 Further Properties of Locally Nilpotent Derivations 35 -- 2.1 Irreducible Derivations 35 -- 2.2 Minimal Local Slices 37 -- 2.3 Three Lemmas about UFDs 39 -- 2.4 Defect of a Derivation 40 -- 2.5 Exponential Automorphisms 44 -- 2.6 Wronskians and Kernel Elements 45 -- 2.7 Star Operator 47 -- 3 Polynomial Rings 49 -- 3.1 Variables, Automorphisms, and Gradings 49 -- 3.2 Derivations of Polynomial Rings 50 -- 3.3 Group Actions on A[superscript n] 61 -- 3.4 Locally Nilpotent Derivations of Polynomial Rings 63 -- 3.5 Slices in Polynomial Rings 66 -- 3.6 Triangular Derivations and Automoprhisms 66 -- 3.7 Homogeneous Locally Nilpotent Derivations 70 -- 3.8 Symmetric Locally Nilpotent Derivations 71 -- 3.9 Some Important Early Examples 72 -- 3.10 Homogeneous Dependence Problem 76 -- 4 Dimension Two 83 -- 4.1 Polynomial Ring in Two Variables over a Field 86 -- 4.2 Locally Nilpotent R-Derivations of R[x, y] 91 -- 4.3 Rank-Two Derivations of Polynomial Rings 97 -- 4.4 Automorphisms Preserving Lattice Points 100 -- 4.5 Newton Polygons 101 -- 4.6 Appendix: Newton Polytopes 103 -- 5 Dimension Three 107 -- 5.1 Miyanishi's Theorem 108 -- 5.2 Other Fundamental Theorems in Dimension Three 115 -- 5.3 Questions of Triangularizability and Tameness 119 -- 5.4 Homogeneous (2, 5) Derivation 121 -- 5.5 Local Slice Constructions 122 -- 5.6 Homogeneous Case 127 -- 5.7 Graph of Kernels and Generalized Local Slice Constructions 131 -- 5.8 G[Characters not reproducible]-Actions 133 -- 5.9 Appendix: An Intersection Condition 134 -- 6 Linear Actions of Unipotent Groups 137 -- 6.1 Finiteness Theorem 138 -- 6.2 Linear G[subscript a]-Actions 139 -- 6.3 Linear Counterexamples to the Fourteenth Problem 146 -- 6.4 Linear G[Characters not reproducible]-Actions 151 -- 6.5 Appendix: Finite Group Actions 155 -- 7 Non-Finitely Generated Kernels 157 -- 7.1 Roberts' Examples 157 -- 7.2 Counterexample in Dimension Five 160 -- 7.3 Proof for A'Campo-Neuen's Example 169 -- 7.4 Quotient of a G[subscript a]-Module 170 -- 7.5 Proof for the Linear Example in Dimension Eleven 173 -- 7.6 Kuroda's Examples in Dimensions Three and Four 174 -- 7.7 Locally Trivial Examples 175 -- 7.8 Some Positive Results 176 -- 7.9 Winkelmann's Theorem 177 -- 7.10 Appendix: Van den Essen's Proof 178 -- 8 Algorithms 181 -- 8.1 Van den Essen's Algorithm 183 -- 8.2 Image Membership Algorithm 185 -- 8.3 Criteria for a Derivation to be Locally Nilpotent 186 -- 8.4 Maubach's Algorithm 188 -- 8.5 Extendibihty Algorithm 190 -- 9 Makar-Limanov and Derksen Invariants 195 -- 9.1 Danielewski Surfaces 197 -- 9.2 A Preliminary Result 199 -- 9.3 Threefold x + x[superscript 2]y + z[superscript 2 ] + t[superscript 3] = 0 201 -- 9.4 Characterizing k[x, y] by LNDs 204 -- 9.5 Characterizing Danielewski Surfaces by LNDs 207 -- 9.6 LNDs of Special Danielewski Surfaces 209 -- 9.7 Further Properties of the ML Invariant 212 -- 9.8 Further Results in the Classification of Surfaces 215 -- 10 Slices, Embeddings and Cancellation 219 -- 10.1 Some Positive Results 220 -- 10.2 Torus Action Formula 223 -- 10.3 Asanuma's Torus Actions 225 -- 10.4 Venereau Polynomials 230 -- 10.5 Open Questions 233 -- 11.1 Rigidity of Kernels for Polynomial Rings 235 -- 11.2 Extension Property 236 -- 11.3 Nilpotency Criterion 236 -- 11.4 Calculating the Makar-Limanov Invariant 236 -- 11.5 Relative Invariants 237 -- 11.6 Structure of LND(B) 237 -- 11.7 Maximal Subalgebras 238 -- 11.8 Invariants of a Sum 238 -- 11.9 Finiteness Problem for Extensions 239 -- 11.10 Geometric Viewpoint 239 -- 11.11 Paragonic Varieties 240 -- 11.12 Stably Triangular G[subscript a]-Actions 241 -- 11.13 Extending G[subscript a]-Actions to Larger Group Actions 241 -- 11.14 Variable Criterion 241 -- 11.15 Bass's Question on Rational Triangularization 242 -- 11.16 Popov's Questions 242 -- 11.17 Miyanishi's Question 242.
Subject Geometry, Algebraic.
Commutative algebra.
Standard Number 9783540295211
ISBN 3540295216 (hd.bd.)