A hypersurface which has the Abhyankar-Moh property
Let \(X\) be an algebraic subset of \(\mathbb{C}^n\). We say that \(X\) has the Abhyankar-Moh property (A.M.P.) if for any polynomial embedding \(f:X\to\mathbb{C}^n\) there exists a polynomial automorphism \(F\) of \(\mathbb{C}^n\) such that \(f\) is a restriction of this automorphism to the set \(X\). It was shown that if \(\dim X\) is sufficiently small relatively to \(n\), and if \(X\) has ``nice'' singularities then \(X\) has the A.M.P. and the problem of extending the polynomial embedding of \(X\) into \(\mathbb{C}^n\) has (in general) many solutions. The analogous questions for a hyperplane and the \(n\)-cross \(K_n= \{x\in\mathbb{C}^n: x_1\cdot\dots\cdot x_n=0\}\) in the \(n\)-space \((n>2)\) has been open (the question for a hyperplane is still open). Moreover, in the case \(n>2\) no example of a hypersurface which had the A.M.P. has been known. In the paper we consider the Abhyankar-Moh property for the \(n\)-cross \[ K_n= \{x\in\mathbb{C}^n: x_1\cdot\dots\cdot x_n=0\} \] in \(\mathbb{C}^n\). The problem whether \(K_n\) has the A.M.P. appears (in an equivalent version) as the ``complementary conjecture'' of McKay and Wang. In the paper we obtain the following affirmative answer: Theorem. For every \(n\geq 1\) the \(n\)-cross \(K_n= \{x\in\mathbb{C}^n: x_1\cdot\dots\cdot x_n=0\}\) has the Abhyankar-Moh property. In particular we give the first example of a hypersurface in dimension \(n>2\) which has the A.M.P.
scientific article; zbMATH DE number 1031500