Locally Nilpotent Derivations over a UFD and an Application to Rank Two Locally Nilpotent Derivations ofk[X1,…,X]

Daigle, Daniel; Freudenburg, Gene

doi:10.1006/jabr.1998.7465

Cited by 49 publications

(45 citation statements)

References 12 publications

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“…This result was first proved by the author and Daigle in [6] for the case R is a UFD. The general case is due to Berson, van den Essen, and Maubach; see [9,Theorem 4.16].…”

Section: Theorem If D Is a Locally Nilpotent R-derivation Of R[x Y supporting

confidence: 51%

Derivations of R[X,Y,Z] with a slice

Freudenburg

2009

Journal of Algebra

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The question under consideration is whether every locally nilpotent R-derivation of R[X, Y, Z] with a slice has kernel A generated by two elements over R, where R is a polynomial ring over a field of characteristic zero. Theorem 1.1 gives a fundamental property of such kernels, namely, that A is an A 2 -fibration over R. While it is an open question whether every A 2 -fibration over R is trivial, the property of A asserted in the theorem is necessary to the condition that A is a polynomial ring in two variables over R. The last section of the paper presents a family of examples θ n (n 1) which are quite simple to define, but whose status relative to the kernel question is not known.

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“…This result was first proved by the author and Daigle in [6] for the case R is a UFD. The general case is due to Berson, van den Essen, and Maubach; see [9,Theorem 4.16].…”

Section: Theorem If D Is a Locally Nilpotent R-derivation Of R[x Y supporting

confidence: 51%

Derivations of R[X,Y,Z] with a slice

Freudenburg

2009

Journal of Algebra

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“…[4]). Let R be a UFD containing Q, let B = R[X, Y ] = R [2] and let K = qt R. For an R-derivation D = 0 of B, the following are equivalent:…”

Section: Derivations Of K[x Y Z]mentioning

confidence: 93%

“…Information on derivations of low rank can be found in references [3,4,7,8]. First, to state some facts which we will need later in our discussion, including the following theorem attributable to Miyanishi [7].…”

Section: Definition 14mentioning

confidence: 98%

Homogeneization of locally nilpotent derivations and an application to k[X,Y,Z]

Wang¹

2005

Journal of Pure and Applied Algebra

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Let k be a field of characteristic zero and let B be a graded k-algebra. We obtain information on a given derivation D : B → B by studying the behavior of the associated homogeneous derivation gr D.As an application, we give a complete classification of locally nilpotent derivations D :We study locally nilpotent derivations of polynomial rings over k, where k is a field of characteristic zero. Our special interest is in so called "nice" derivations over a polynomial ring k[X 1 , . . . , X n ], which satisfies D 2 X i = 0 for all i. The main result of the paper is the following.Y, Z] and let 0 = D : B → B be a locally nilpotent derivation. If D is irreducible and D 2 X = 0 = D 2 Y , then one of the following holds: 1. There exists a coordinate system (L 1 , L 2 , Z) of B, where L 1 and L 2 are linear forms in X, Y , such that D(L 1 ) = 0, D(L 2 ) ∈ k[L 1 ] and D(Z) ∈ k[L 1 , L 2 ] = k[X, Y ]. *

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“…By Lemma 2, we deduce that [7]), Bhatwadekar and Dutta (for the case R normal noetherian domain, see [6]), Berson, van den Essen and Maubach (for the general case, see [5]): Let R be a Q-algebra and let F ∈ R[x, y] be a polynomial. We have: F ∈ VA 2 (R) if and only if F ∈ VA 2 (qt(R)) and (…”

Section: )mentioning

confidence: 99%

Coordinates ofR[x,y]: Constructions and Classifications

Edo¹

2013

Communications in Algebra

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Abstract. Let R be a PID. We construct and classify all coordinates of R [x, y] of the form p 2 y + Q 2 (p 1 x + Q 1 (y)) with p 1 , p 2 ∈ qt(R) and Q 1 , Q 2 ∈ qt(R) [y]. From this construction (with R = K[z]) we obtain non tame automorphisms σ of K[x, y, z] (where K is a field of characteristic 0) such that the sub-group generated by σ and the affine automorphisms contains all tame automorphisms.

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Locally Nilpotent Derivations over a UFD and an Application to Rank Two Locally Nilpotent Derivations ofk[X1,…,X]

Cited by 49 publications

References 12 publications

Derivations of R[X,Y,Z] with a slice

Derivations of R[X,Y,Z] with a slice

Homogeneization of locally nilpotent derivations and an application to k[X,Y,Z]

Coordinates ofR[x,y]: Constructions and Classifications

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