A generalization of the Shestakov-Umirbaev inequality

Kuroda, Shigeru

doi:10.2969/jmsj/06020495

Cited by 17 publications

(27 citation statements)

References 4 publications

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“…The inequality of Shestakov and Umirbaev used in this proof has recently been generalized by both Kuroda [14] and Vénéreau [21]. Each obtains the original inequality as a corollary, avoiding the * -reduced pair condition used by Shestakov and Umirbaev.…”

Section: Applying the Inequality Of Shestakov And Umirbaevmentioning

confidence: 89%

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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Section: Applying the Inequality Of Shestakov And Umirbaevmentioning

confidence: 89%

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

Freudenburg

2009

Journal of Algebra

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“…Then, due to [9, Theorem 3] (see also [5], [7] and [11]), it follows that Recently, the author [6] showed that no tame automorphism of k[x] for n = 3 admits a reduction of type IV.…”

Section: Question 41mentioning

confidence: 99%

Automorphisms of a Polynomial Ring Which Admit Reductions of Type I

Kuroda¹

2009

Publ. Res. Inst. Math. Sci.

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Recently, Shestakov-Umirbaev solved Nagata's conjecture on an automorphism of a polynomial ring. To solve the conjecture, they defined notions called reductions of types I-IV for automorphisms of a polynomial ring. An automorphism admitting a reduction of type I was first found by Shestakov-Umirbaev. Using a computer, van den Essen-Makar-Limanov-Willems gave a family of such automorphisms. In this paper, we present a new construction of such automorphisms using locally nilpotent derivations. As a consequence, we discover that there exists an automorphism admitting a reduction of type I which satisfies some degree condition for each possible value.

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“…To prove Theorem 1.9, we use the generalized Shestakov-Umirbaev theory [12], [13]. For the convenience of the reader, we give a short introduction to this theory.…”

Section: Shestakov-umirbaev Reductionsmentioning

confidence: 99%

“…By definition, we have deg w f 3 = w 3 and f w 3 = αx 3 + p ′ for such F , where p ′ := p w if deg w p = w 3 , and p ′ := 0 otherwise. Recently, the author [12], [13] generalized the Shestakov-Umirbaev theory. By means of this theory, we prove the following theorem in Section 9.…”

Section: Introductionmentioning

confidence: 99%