On simple Shamsuddin derivations in two variables

Baltazar, Rene

doi:10.1590/0001-3765201620140718

Cited by 9 publications

(3 citation statements)

References 11 publications

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“…Based on Baltazar ([1], [2]) and Mendes-Pan results ( [12]) and also on the results and examples we presented in this paper, we can make the following conjecture:…”

mentioning

confidence: 65%

On the isotropy group of a simple derivation

Bertoncello

Levcovitz

2020

Journal of Pure and Applied Algebra

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Let R = K[X 1 , . . . , Xn] be a polynomial ring in n variables over a field K of charactersitic zero and d a K-derivation of R. Consider the isotropy group if d: Aut(R) d := {ρ ∈ Aut K (R)| ρdρ −1 = d}. In his doctoral thesis ([1]), Baltazar proved that if d is a simple Shamsuddin derivation of K[X 1 , X 2 ], then its isotropy group is trivial. He also gave an example of a non-simple derivation whose isotropy group is infinite. Recently, Mendes and Pan ([12]) generalized this result to an arbitrary derivation of K[X 1 , X 2 ] proving that a derivation of K[X 1 , X 2 ] is simple if, and only if, its isotropy group is trivial. In this paper, we prove that the isotropy group of a simple Shamsuddin derivation of the polynomial ring R = K[X 1 , . . . , Xn] is trivial. We also calculate other isotropy groups of (not necessarily simple) derivations of K[X 1 , X 2 ] and prove that they are finite cyclic groups.

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“…Based on Baltazar ([1], [2]) and Mendes-Pan results ( [12]) and also on the results and examples we presented in this paper, we can make the following conjecture:…”

mentioning

confidence: 65%

On the isotropy group of a simple derivation

Bertoncello

Levcovitz

2020

Journal of Pure and Applied Algebra

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“…. , x n ) for all 1 ≤ i ≤ n. Then we have ρ(m) = m. It follows from Proposition 7 in [3] that ρ = id. That is, S = {id}.…”

Section: Introductionmentioning

confidence: 85%

“…Rene Baltazar shows that Conjecture 1.1 is true for the Shamsuddin derivations of K[x 1 , x 2 ] in [3]. In particular, the author proves that Aut…”

Section: Introductionmentioning

confidence: 95%

On simple derivations and the group of polynomial automorphisms commuting with certain derivations

Yan¹

2018

Preprint

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In the paper, we first give an equivalent statement about Aut(K[x 1 , . . . ,x n ]) D = {id} and some properties of D if D is a simple derivation. Then we study the subgroup of K-automorphisms of K[x 1 , . . . , x n ] which commutes with simple derivations of K[x 1 , . . . , x n ] base on the equivalent statement.We show that the subgroup of K-automorphisms of K[x 1 , x 2 ] which commutes with simple derivations is trivial in the second section. Finally, we show that the subgroup of K-automorphisms of K[x 1 , . . . , x n ] which commutes with simple Shamsuddin derivations is trivial.

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