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Extensions birationnelles simples de l'anneau de polynômes ℂ³ = Simple birational extensions of the polynomial ring ℂ³
Journal
Comptes rendus de l'Académie des sciences. Série 1, Mathématique. Series 1, Mathematics
Volume
333, H. 4
Number
4
Pages / Article-Number
319-322
Abstract
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.
