Algebraic Theory of Locally Nilpotent Derivations

Abstract

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. 1. First principles -- 2. Further properties of locally Nilpotent derivations -- 3. Polynomial rings -- 4. Dimension two -- 5. Dimension three -- 6. Linear actions of unipotent groups -- 7. Non-finitely generated kernels -- 8. Algorithms -- 9. The Makar-Limanov and Derksen invariants -- 10. Slices, embeddings, and cancellation -- 11. Epilogue.