By Gene Freudenburg

This ebook explores the idea and alertness of in the neighborhood nilpotent derivations, that's a subject matter of turning out to be curiosity and significance not just between these in commutative algebra and algebraic geometry, but additionally in fields reminiscent of Lie algebras and differential equations. the writer presents a unified therapy of the topic, starting with sixteen First ideas on which the full thought is predicated. those are used to set up classical effects, corresponding to Rentschler's Theorem for the airplane, correct as much as the newest effects, corresponding to Makar-Limanov's Theorem for in the neighborhood nilpotent derivations of polynomial earrings. subject matters of unique curiosity comprise: growth within the size 3 case, finiteness questions (Hilbert's 14th Problem), algorithms, the Makar-Limanov invariant, and connections to the Cancellation challenge and the Embedding challenge. The reader also will discover a wealth of pertinent examples and open difficulties and an updated source for study.

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**Example text**

Conjugate to a linear derivation. , xn ). We say that D ∈ Derk (B) is quasi-linear if and only if there exists a matrix M ∈ Mn (ker D) such that D(x) = M x. ) Then D is locally nilpotent if and only if M is a nilpotent matrix. , n and Dx1 ∈ k. Note that triangularity depends on the choice of coordinates on B. , conjugate to a triangular derivation. As we will see, the triangular derivations form a very large and important class of locally nilpotent derivations of polynomial rings. Several of the main examples and open questions discussed below involve triangular derivations.

If r ∈ min(D), then r = as + b for some s ∈ min(D) and some non-unit a ∈ A. Thus, Dr = a · Ds, and since Dr is irreducibile, Ds ∈ B ∗ . 7. Given D ∈ LND(B) with A = ker D, the following are equivalent. 1. 2. D has a unique minimal local slice (up to equivalence). The plinth ideal DB ∩ A is a principal ideal of A. Proof. Suppose that D has only one minimal local slice r ∈ B, up to equivalence, and let a ∈ DB ∩ A be given. Then there exists a local slice p of D such that Dp = a. By hypothesis, there exist b, c ∈ A with p = br + c.

Proof. By induction, it suﬃces to show that B [1] has the ACC on principal ideals. Suppose (p1 (t)) ⊂ (p2 (t)) ⊂ (p3 (t)) ⊂ · · · is an inﬁnite chain of of principal ideals, where pi (t) ∈ B[t] (t an indeterminate over B). Since B is a domain, the degrees of the pi (t) must stabilize, so we may assume (truncating the chain if necessary) that for some positive integer d, degt pi (t) = d for all i. ) Thus, given i, there exist ei ∈ B with pi (t) = ei pi+1 (t). For each integer m with 0 ≤ m ≤ d, (m) let pi denote the coeﬃcient of tm in pi (t).