By Gene Freudenburg

This booklet explores the speculation and alertness of in the neighborhood nilpotent derivations, that's an issue of starting to be curiosity and significance not just between these in commutative algebra and algebraic geometry, but in addition in fields resembling Lie algebras and differential equations. the writer offers a unified therapy of the topic, starting with sixteen First ideas on which the full idea relies. those are used to set up classical effects, corresponding to Rentschler's Theorem for the airplane, correct as much as the newest effects, equivalent to Makar-Limanov's Theorem for in the neighborhood nilpotent derivations of polynomial earrings. issues of distinctive curiosity comprise: growth within the measurement 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 examine.

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**Example text**

If Df ∈ f B for D ∈ LND(B) and f ∈ B, then Df = 0. Principle 6. (Prop. 1 of [116]) Suppose B is a commutative k-domain, and D ∈ LND(B). Assume that D is extended to a derivation D∗ of the ring B[t] = B [1] . Then D∗ is locally nilpotent if and only if D∗ t ∈ B. Proof. If D∗ t ∈ B, then since B ⊂ Nil(D∗ ), it follows that t ∈ Nil(D∗ ). So in this case, D∗ is locally nilpotent by Princ. 2 above. Conversely, assume D∗ is locally nilpotent, but that D∗ t ∈ B. Choose N such that (D∗ )N t ∈ B, but (D∗ )N +1 t ∈ B, which is possible since D∗ is locally nilpotent.

Then D = 0. Set N = νf D (f ) ≥ 0, and choose g ∈ B − Nil(D). It follows that g = 0, νf D (g) ≥ 0, and νf D (Dn g) ≥ 0 for all n ≥ 1. On the one hand, we have νf D (f · Dn g) = νf D ((f D)(Dn−1 g)) = νf D (Dn−1 g) − 1 . On the other hand, we see that νf D (f · Dn g) = νf D (f ) + νf D (Dn g) = N + νf D (Dn g) . Therefore, νf D (Dn g) = νf D (Dn−1 g) − (N + 1) for all n ≥ 1 . 4 First Principles for Locally Nilpotent Derivations 25 This implies νf D (Dn g) = νf D (g) − n(N + 1) , which is absurd since it means νf D has values in the negative integers.

2) holds, and δ is a derivation. = aδb + bδa 32 1 First Principles To see that δ is locally nilpotent, let f ∈ B be given, and suppose ρ∗ (f ) = P (t) = 0≤i≤n fi ti for fi ∈ B. For general s, t ∈ k, we have (s + t) · f = s · (t · f ) = 0≤i≤n (s · fi )ti . On the other hand, it follows from Taylor’s formula that (s + t) · f = P (s + t) = 0≤i≤n P (i) (s) i t . i! )P (i) (s) for all s ∈ k. We now proceed by induction on the t-degree of ρ∗ (f ). If the degree is zero, then δ(f ) = P ′ (0) = 0, and thus f ∈ Nil(δ).