Algebraic Theory of Locally Nilpotent Derivations

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.