Simple Birational Extensions of the Polynomial Ring
نویسنده
چکیده
The Abhyankar-Sathaye Problem asks whether any biregular embedding φ : C →֒ C can be rectified, that is, whether there exists an automorphism α ∈ AutC such that α ◦ φ is a linear embedding. Here we study this problem for the embeddings φ : C3 →֒ C4 whose image X = φ(C3) is given in C4 by an equation p = f(x, y)u + g(x, y, z) = 0, where f ∈ C[x, y]\{0} and g ∈ C[x, y, z]. 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., a coordinate of a polynomial automorphism of C4). This is an analog of a theorem due to Sathaye [30] which concerns the case of embeddings C2 →֒ C3. Besides, we generalize a theorem of Miyanishi [24, Thm. 2] giving, for a polynomial p as above, a criterion for as when X = p−1(0) ≃ C3.
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تاریخ انتشار 1999