A hypersurface which has the Abhyankar-Moh property
type: Item
instance of: scholarly article
title: A hypersurface which has the Abhyankar-Moh property
zbMATH Open document ID: 0870.14004
author: Z. Jelonek
published in: Mathematische Annalen
publication date: 1997-09-17T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 14E05
Mathematics Subject Classification ID: 14H37
Mathematics Subject Classification ID: 14J70
zbMATH DE Number: 1031500
zbMATH Keywords: Abhyankar-Moh property
zbMATH Keywords: polynomial automorphism
zbMATH Keywords: extending the polynomial embedding
zbMATH Keywords: \(n\)-cross
MaRDI profile type: Publication
full work available at URL: s002080050065
OpenAlex ID: W1991694864
DOI: 10.1007/S002080050065
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description: scientific article; zbMATH DE number 1031500