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A Remark on a Paper of Crachiola and Makar-Limanov

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Dryło R.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2011
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tom 59
|
nr 3
203-206
EN
A. Crachiola and L. Makar-Limanov [J. Algebra 284 (2005)] showed the following: if X is an affine curve which is not isomorphic to the affine line $𝔸¹_k$, then ML(X×Y) = k[X]⊗ ML(Y) for every affine variety Y, where k is an algebraically closed field. In this note we give a simple geometric proof of a more general fact that this property holds for every affine variety X whose set of regular points is not k-uniruled.
2
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Non-uniruledness and the cancellation problem (II)

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Dryło R.
Annales Polonici Mathematici
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2007
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tom 92
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nr 1
41-48
EN
We study the following cancellation problem over an algebraically closed field 𝕂 of characteristic zero. Let X, Y be affine varieties such that $X × 𝕂^m ≅ Y × 𝕂^m$ for some m. Assume that X is non-uniruled at infinity. Does it follow that X ≅ Y? We prove a result implying the affirmative answer in case X is either unirational or an algebraic line bundle. However, the general answer is negative and we give as a counterexample some affine surfaces.
3
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Non-uniruledness and the cancellation problem

100%
Dryło R.
Annales Polonici Mathematici
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2005
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tom 87
|
nr 1
93-98
EN
Using the notion of uniruledness we indicate a class of algebraic varieties which have a stronger version of the cancellation property. Moreover, we give an affirmative solution to the stable equivalence problem for non-uniruled hypersurfaces.
4
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On the stable equivalence problem for k[x,y]

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Dryło R.
Colloquium Mathematicum
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2011
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tom 124
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nr 2
247-253
EN
L. Makar-Limanov, P. van Rossum, V. Shpilrain and J.-T. Yu solved the stable equivalence problem for the polynomial ring k[x,y] when k is a field of characteristic 0. In this note we give an affirmative solution for an arbitrary field k.
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