Simple birational extensions of the polynomial algebra ℂ³

Kaliman, Shulim and Vénéreau, Stéphane and Zaidenberg, Mihail. (2004) Simple birational extensions of the polynomial algebra ℂ³. Transactions of the American Mathematical Society, Vol. 356, H. 2. pp. 509-555.

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Official URL: http://edoc.unibas.ch/dok/A5253592

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The Abhyankar-Sathaye Problem asks wether any biregular embeddingof Ck into Cn can be rectified. Here we studythis problem for the embeddings of C3 into C4 whose image X isgiven in C4 by an equation p = f(x; y)u+g(x; y; z) = 0. Under certain additional assumptions we show that, indeed,the polynomial p is a variable of the polynomial ring C[4] = C[x; y; z; u] (i.e., acoordinate of a polynomial automorphism of C4). This is an analog of a theoremdue to Sathaye (1976) which concerns the case of embeddings of C2 into C3.Besides, we generalize a theorem of Miyanishi (1984) giving, for a polynomialp as above, a criterion for when X is isomorphic to C3.
Faculties and Departments:05 Faculty of Science > Departement Mathematik und Informatik > Ehemalige Einheiten Mathematik & Informatik > Algebra (Kraft)
UniBasel Contributors:Vénéreau, Stéphane
Item Type:Article, refereed
Article Subtype:Research Article
Publisher:Johns Hopkins Univ. Press
Note:Publication type according to Uni Basel Research Database: Journal article
Last Modified:22 Mar 2012 14:27
Deposited On:22 Mar 2012 13:58

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