New bad lines in \(R[x,y]\) and optimization of the epimorphism theorem
The article under review studies conditions on a ring \(R\) under which all lines in \(R[x,y]\) are variables. A polynomial \(f(x,y)\in R[x,y]\) is called a line or an embedded line if \(R[x,y]/(f)\cong R[t]\). It is called a variable or a coordinate if there exists a \(g\in R[x,y]\) so that \(R[f,g]=R[x,y]\). If \(R\) is a field of characteristic zero, all lines are variables was proved by \textit{S. S. Abhyankar} and \textit{T. Moh} [J. Reine Angew. Math. 276, 148--166 (1975; Zbl 0332.14004)]. The characteristic zero hypothesis is essential and there are positive characteristic counterexamples due to \textit{M. Nagata} [J. Math. Kyoto Univ. 11, 149--154 (1971; Zbl 0213.23201)]. It was known that there are examples of \(R\) over which lines are not variables even if characteristic is zero. In this paper the author gives necessary and sufficient conditions on \(R\) under which any line in \(R[x,y]\) is a variable. The result involves a generalization of seminormality, which the author calls \(n\)-seminormality.
scientific article; zbMATH DE number 5078145