Algebraic Characterization of the Affine Plane and the Affine 3-Space

Sugie, Toru

doi:10.1007/978-1-4612-3702-0_12

Cited by 10 publications

(4 citation statements)

References 10 publications

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“…Now let's return to the cancellation problems described above. (For more details we refer to the paper [36] by Kraft, the papers [33] and [34] by Kang, the paper [57] by Sugie and the paper [45] by Miyanishi and Sugie. )…”

Section: Cancellation Problemsmentioning

confidence: 99%

Polynomial automorphisms and the Jacobian conjecture, by Arno van den Essen. Progress in Mathematics 190. Pp. 352. SFr 98. 2000. ISBN 3 7643 6350 9 (Birkhäuser).

Abbott¹

2001

Math. Gaz.

142

258

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In this paper we give an update survey of the most important results concerning the Jacobian conjecture: several equivalent descriptions are given and various related conjectures are discussed. At the end of the paper, we discuss the recent counter-examples, in all dimensions greater than two, to the Markus-Yamabe conjecture (Global asymptotic Jacobian conjecture). RésuméDans ce papier nous présentons un rapport actualisé sur les résultats les plus importants concernant la conjecture Jacobienne : plusieurs formulations équivalentes et diverses conjectures connexes sont considérées. A la fin du papier, nous donnons les contre-exemples récents, en toute dimension plus grande que deux, à la conjecture de Markus-Yamabe.

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Section: Cancellation Problemsmentioning

confidence: 99%

Polynomial automorphisms and the Jacobian conjecture, by Arno van den Essen. Progress in Mathematics 190. Pp. 352. SFr 98. 2000. ISBN 3 7643 6350 9 (Birkhäuser).

Abbott¹

2001

Math. Gaz.

142

258

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“…, s n−1 ] and X i = h i (c)∂ s i is false in general. We show below that C[x, y, z, t]/f , where f = x + x 2 y + z 3 + t 2 , gives a counterexample to this (this example is used to show that Miyanishi and Sugie characterization of the affine plane [13,17] does not hold in higher dimensions).…”

Section: Proof the Canonical Projection Frommentioning

confidence: 98%

UFDs with commuting linearly independent locally nilpotent derivations

Kahoui

2005

Journal of Algebra

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In this paper we study affine K-UFDs of transcendence degree n without nonconstant units, having n − 1 commuting linearly independent locally nilpotent K-derivations. We prove in case n = 2, and K algebraically closed of characteristic zero, that such rings are polynomial rings in two variables over K. We then show that the commuting derivations Conjecture is equivalent to a weak version of the Abhyankar-Sathaye Conjecture.  2005 Elsevier Inc. All rights reserved.

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“…The cancellation problem asks for X = C n whether a G a action which is conjugate to a translation has Y ∼ = C n−1 . The cancellation problem is known to have a positive solution only for n ≤ 3 (see [16] for instance).…”

Section: Introductionmentioning

confidence: 99%

Free $G_a$ actions on $C^3$

Finston¹,

Deveney²

1999

Proc. Amer. Math. Soc.

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Abstract. It has been conjectured that every free algebraic action of the additive group of complex numbers on complex affine three space is conjugate to a global translation. The main result lends support to this conjecture by showing that the morphism to the variety defined by the ring of invariants of the associated action on the coordinate ring is smooth. As a consequence, the graph morphism is an open immersion, and simple proofs of certain cases of the conjecture are obtained.

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Algebraic Characterization of the Affine Plane and the Affine 3-Space

Cited by 10 publications

References 10 publications

Polynomial automorphisms and the Jacobian conjecture, by Arno van den Essen. Progress in Mathematics 190. Pp. 352. SFr 98. 2000. ISBN 3 7643 6350 9 (Birkhäuser).

Polynomial automorphisms and the Jacobian conjecture, by Arno van den Essen. Progress in Mathematics 190. Pp. 352. SFr 98. 2000. ISBN 3 7643 6350 9 (Birkhäuser).

UFDs with commuting linearly independent locally nilpotent derivations

Free $G_a$ actions on $C^3$

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