Simple birational extensions of the polynomial ring \(\mathbb{C}^{[3]}\) (Q2748290)

This note contains results related to the Abhyankar-Sathaye problem in the case of embeddings \(\phi:\mathbb{C}^3\hookrightarrow \mathbb{C}^4\) where the image is given by a polynomial of type \(p=f(x,y)u+g(x,y,z)\). The authors show that under certain assumptions such an embedding can be rectified (a result that has been achieved for embeddings \(\mathbb{C}^2\hookrightarrow \mathbb{C}^3\) by \textit{A. Sathaye} [Proc. Am. Math. Soc. 56, 1--7 (1976; Zbl 0345.14013]). Furthermore, several equivalent conditions for the existence of an isomorphism \(p^{-1}(0)\simeq \mathbb{C}^3\) are given, generalizing a theorem of \textit{M. Miyanishi} [Am. J. Math 106, 1469--1485 (1984; Zbl 0595.14025)].NEWLINENEWLINEMeanwhile, the authors published a comprehensive article on this subject [Trans. Am. Math. Soc. 356, No.2, 509--555 (2004; Zbl 1041.14026)].