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- W1597946181 abstract "AbstractThis paper studies polynomials of the form p(X)Y + q(X,Z 1 ,...,Z n−1 ),mainly on the question when such a polynomial is a coordinate. It is shownthat a polynomial of the form p(X)Y + Q(X,Z,T) is a coordinate if and onlyif Q(a,Z,T) is a coordinate for every zero a of p(X). Connections with theCancellation Problem are made. As a consequence, the Abhyankar-SathayeConjecture has a positive answer for polynomials in four variables of this form.Also, conjectures about possible generalisations of the concept of “coordinate”for elements of general rings are made. 1 Introduction In his Bourbaki lecture ([11]) Kraft gave a list of various challenging prob-lems in affine geometry. It can be shown that most of these problemsare related to the following crucial question: given a polynomial f inan n-variable polynomial ring over C, denoted C [n] , how can one recog-nise if the zero-set of f is isomorphic to affine (n − 1)-space over C, orequivalently if C [n] /(f) is a polynomial ring in n − 1 variables over C?If n = 2 this problem was solved by Abhyankar and Moh in [1]: theyshowed that C" @default.
- W1597946181 created "2016-06-24" @default.
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- W1597946181 date "2001-01-01" @default.
- W1597946181 modified "2023-09-27" @default.
- W1597946181 title "ON THE ABHYANKAR-SATHAYE CONJECTURE IN FOUR VARIABLES" @default.
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