Simple birational extensions of the polynomial ring C-[3]

The Abhyankar-Sathaye problem asks whether any biregular embedding phi :C(k )hooked right arrowC(n) can be rectified, that is, whether there exists an automorphism alpha is an element of Aut C-n such that alpha circle phi is a linear embedding. In the spit-it of [5], here we study this problem for th e embeddings phi: C(3)hooked right arrowC(4) whose image X = phi (C-3) is g iven in C-4 by an equation p = f (x, y)u + g(x, y, z) = 0, where f is an el ement of C[x, y]\{0} and g is an element of C[x, y, z]. Under certain addit ional assumptions we show that, indeed, the polynomial p is a variable of t he polynomial ring C-[4] = C[x, y, z, u] (i.e., a coordinate of a polynomia l automorphism of C-4). For that, we first study the acyclicity of X in a m ore general setting. Then we give several equivalent conditions for X = p(- 1)(0) similar or equal toC(3) generalizing in particular a theorem of Miyan ishi [4, Thm. 2]. (C) 2001 Academie des sciences/Editions scientifiques et medicales Elsevier SAS.