© 2006

Algebraic Theory of Locally Nilpotent Derivations


Part of the Encyclopaedia of Mathematical Sciences book series (EMS, volume 136)

Table of contents

  1. Front Matter
    Pages I-XI
  2. Gene Freudenburg
    Pages 1-7
  3. Gene Freudenburg
    Pages 9-33
  4. Gene Freudenburg
    Pages 49-82
  5. Gene Freudenburg
    Pages 83-106
  6. Gene Freudenburg
    Pages 107-136
  7. Gene Freudenburg
    Pages 137-156
  8. Gene Freudenburg
    Pages 157-180
  9. Gene Freudenburg
    Pages 181-194
  10. Gene Freudenburg
    Pages 195-217
  11. Gene Freudenburg
    Pages 219-234
  12. Gene Freudenburg
    Pages 235-242
  13. Back Matter
    Pages 243-261

About this book


This book explores the theory and application of locally nilpotent derivations, which is a subject of growing interest and importance not only among those in commutative algebra and algebraic geometry, but also in fields such as Lie algebras and differential equations. 

The author provides a unified treatment of the subject, beginning with 16 First Principles on which the entire theory is based. These are used to establish classical results, such as Rentschler’s Theorem for the plane, right up to the most recent results, such as Makar-Limanov’s Theorem for locally nilpotent derivations of polynomial rings. Topics of special interest include: progress in the dimension three case, finiteness questions (Hilbert’s 14th Problem), algorithms, the Makar-Limanov invariant, and connections to the Cancellation Problem and the Embedding Problem. The reader will also find a wealth of pertinent examples and open problems and an up-to-date resource for research.  


Dimension additive group action on affine varieties algebra algebraic geometry commutative algebra invariant theory locally nilpotent derivation

Authors and affiliations

  1. 1.Department of MathematicsWestern Michigan UniversityKalamazooUSA

Bibliographic information


From the reviews:

"In the volume under review, the author gives a detailed description of the subject covering all the important results … . the book has a wealth of examples and the Epilogue details some important open problems in the area. … is accessible to less advanced graduate students. It is a valuable addition to the literature and am sure would be very helpful to the interested student and researcher alike." (N. Mohan Kumar, Zentralblatt MATH, Vol. 1121 (23), 2007)