Algebraic Theory of Locally Nilpotent Derivations

  • Gene¬†Freudenburg

Part of the Encyclopaedia of Mathematical Sciences book series (EMS, volume 136.3)

Table of contents

  1. Front Matter
    Pages i-xxii
  2. Gene Freudenburg
    Pages 1-39
  3. Gene Freudenburg
    Pages 41-72
  4. Gene Freudenburg
    Pages 73-112
  5. Gene Freudenburg
    Pages 113-136
  6. Gene Freudenburg
    Pages 137-165
  7. Gene Freudenburg
    Pages 167-191
  8. Gene Freudenburg
    Pages 193-216
  9. Gene Freudenburg
    Pages 217-243
  10. Gene Freudenburg
    Pages 245-264
  11. Gene Freudenburg
    Pages 265-285
  12. Gene Freudenburg
    Pages 287-298
  13. Back Matter
    Pages 299-319

About this book


This book explores the theory and application of locally nilpotent derivations, a subject motivated by questions in affine algebraic geometry and having fundamental connections to areas such as commutative algebra, representation theory, Lie algebras and differential equations.

The author provides a unified treatment of the subject, beginning with 16 First Principles on which the theory is based. These are used to establish classical results, such as Rentschler's Theorem for the plane and the Cancellation Theorem for Curves.

More recent results, such as Makar-Limanov's theorem for locally nilpotent derivations of polynomial rings, are also discussed. Topics of special interest include progress in classifying additive actions on three-dimensional affine space, finiteness questions (Hilbert's 14th Problem), algorithms, the Makar-Limanov invariant, and connections to the Cancellation Problem and the Embedding Problem.

A lot of new material is included in this expanded second edition, such as canonical factorization of quotient morphisms, and a more extended treatment of linear actions. The reader will also find a wealth of examples and open problems and an updated resource for future investigations.


dimension additive group action on affine varieties algebra algebraic geometry commutative algebra invariant theory

Authors and affiliations

  • Gene¬†Freudenburg
    • 1
  1. 1.Department of MathematicsWestern Michigan UniversityKalamazooUSA

Bibliographic information

  • DOI
  • Copyright Information Springer-Verlag GmbH Germany 2017
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-662-55348-0
  • Online ISBN 978-3-662-55350-3
  • Series Print ISSN 0938-0396
  • Buy this book on publisher's site