By Gene Freudenburg
This booklet explores the speculation and alertness of in the neighborhood nilpotent derivations, that's an issue of becoming curiosity and value not just between these in commutative algebra and algebraic geometry, but additionally in fields equivalent to Lie algebras and differential equations. the writer presents a unified remedy of the topic, starting with sixteen First rules on which the full conception relies. those are used to set up classical effects, similar to Rentschler's Theorem for the airplane, correct as much as the newest effects, comparable to Makar-Limanov's Theorem for in the neighborhood nilpotent derivations of polynomial earrings. subject matters of detailed 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 up to date source for study.
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Extra info for Algebraic Theory of Locally Nilpotent Derivations
B) If E is locally nilpotent and D = log(I + E), then D ∈ LND(B) and f = exp D. Of course, there may be simpler criteria showing that an automorphism ϕ is not an exponential automorphism. For example, exp D cannot have ﬁnite order when D = 0, since (exp D)n = exp(nD). 6 Wronskians and Kernel Elements In this section, Wronskian determinants associated to a derivation are deﬁned, and some of their basic properties are given. They are especially useful for constructing constants for derivations. The proofs for this section are elementary, and most are left to the reader.
If D ∈ LND(B) has local slice r ∈ B, set A = ker D and f = Dr. Then Bf = Af [r] and the extension d . Thus, the induced Ga -action exp(tD) on X = SpecB of D to Bf equals dr restricts to an equivariantly trivial action on the principal open set Uf deﬁned by f . , fn ∈ pl(D) = A∩DB and satisfy f1 B +· · ·+fn B = B, then the principal open sets Ufi cover X, and the Ga -action on X is locally trivial (hence ﬁxed-point free) relative to these open sets. And ﬁnally, if D admits a slice, then X = Y × A1 for Y = SpecA, and the action of Ga on X is equivariantly trivial relative to this decomposition: t · (y, z) = (y, z + t).
5 Ga -Actions In this section, assume that the ﬁeld k is algebraically closed (still of characteristic zero). Let B be an aﬃne k-domain, and let X = Spec(B) be the corresponding aﬃne variety. Given D ∈ LND(B), by combining Princ. 7 and Princ. 10, we obtain a group homomorphism η : (ker D, +) → Autk (B) , η(f ) = exp(f D) . In addition, if D = 0, then η is injective. Restricting η to the subgroup Ga = (k, +), we obtain the algebraic representation η : Ga ֒→ Autk (B). Geometrically, this means that D induces the faithful algebraic Ga -action exp(tD) on X (t ∈ k).
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