Stably tame coordinates
For a commutative ring \(A\), the author defines a class of polynomials \({\mathcal B}(A)\subset A[x,y]\), the polynomial ring in 2 variables over \(A\). Elements of \({\mathcal B}(A)\) are defined in nested sequences \(\{ F_n\}_{n\geq 0}\), where \(F_n=aF_{n-2}+P(F_{n-1})\) for \(a\in A\) and \(P\in A[y]\), and where the initial pair \((F_0,F_1)\) is triangular (see Definition 1.6). The main questions are: (1) Which \(F\in {\mathcal B}(A)\) are \(A\)-variables of \(A[x,y]\); (2) which \(F\in {\mathcal B}(A)\) are tame \(A\)-variables of \(A[x,y]\); and (3) which \(F\in {\mathcal B}(A)\) are \(A\)-variables of \(A[x,y,z_1,\dots,z_m]\) for some \(m\)? Of course, the answers depend on the base ring \(A\). When \(A\) is a field, results are known classically. The author produces several new results for general rings \(A\), the main of which is Theorem 3.1: If \(F\in {\mathcal B}(A)\) is an \(A\)-variable of \(A[x,y]\), then it is a tame \(A\)-variable of \(A[x,y,z_1,\dots,z_m]\) for some \(m\geq 0\).
scientific article; zbMATH DE number 1758357