Abhyankar-Sathaye embedding problem in dimension three (Q1425553)

Let \(f\in\mathbb{C}[x_1,\dots,x_n]\) be an irreducible polynomial in \(n\) complex variables. Suppose that the hypersurface \(S:=(f=0)\) in the affine space \(\mathbb{A}^n\) is isomorphic to \(\mathbb{A}^{n-1}\). The Abhyankar-Sathaye embedding problem asks whether the polynomial \(f\) is a variable of \(\mathbb{C}[x_1,\dots,x_n]\); i.e., there exist polynomials \(f_1=f, f_2,\dots, f_n\in\mathbb{C}[x_1,\dots,x_n]\) such that \(\mathbb{C}[f_1,\dots,f_n]=\mathbb{C}[x_1,\dots,x_n]\). A polynomial \(f\) is a variable if and only if there is an automorphism \(\phi\) of the algebra \(\mathbb{C}[x_1,\dots,x_n]\) such that \(\phi(x_1)=f\). Hence it is an important problem to determine the form of such polynomials. In the case \(n=2\) the affirmative answer to the Abhyankar-Sathaye problem was given independently by \textit{S. S. Abhyankar} and \textit{T. T. Moh} [J. Reine Angew. Math. 276, 148--166 (1975; Zbl 0332.14004)] and \textit{M. Suzuki} [J. Math. Soc. Japan 26, 241--257 (1974; Zbl 0276.14001)]. Meanwhile, in the case \(n\geq 3\) the problem is still open. Some positive results in this direction were obtained by Sathaye, Russell, Miyanishi, Wright and Ohta. The aim of the present paper is to obtain the positive solution of the Abhyankar-Sathaye problem for \(n=3\) under some restrictions. Namely, suppose that \(\deg(f)=d\geq 2\) and embed \(\mathbb{A}^3\) into the projective space \(\mathbb{P}^3\) canonically as the complement of the hyperplane \(H_0:=(x_0=0)\), where \((x_0:x_1:x_2:x_3)\) are the homogeneous coordinates of \(\mathbb{P}^3\). Let \(X\) be the closure of the hypersurface \(S\subset\mathbb{A}^3\) in \(\mathbb{P}^3\) and \(L\) the (set-theoretic) intersection of \(X\) and \(H_0\). The author considers the case where \(L\) is a line and the hypersurface \(X\) has multiplicity \((d-1)\) along the line \(L\). In order to prove the result, consider the polynomial map \(\mathbb{A}^3\to\mathbb{A}^1\) given by \((x_1,x_2,x_3)\to f(x_1,x_2,x_3)\). It is shown in the paper under review that all fibers of this map are isomorphic to \(\mathbb{A}^2\) and that this map gives a trivial \(\mathbb{A}^2\)-bundle structure on \(\mathbb{A}^3\) over the base curve \(\mathbb{A}^1\). The author also obtains all possible forms of the polynomials satisfying the above conditions.