Finitely generated algebras associated with rational vector fields (Q1375961)

Let \(k\) be an algebraically closed field of characteristic zero and \(R=k[x,y]\). Let \(\delta={1\over f}{\partial\over\partial x}+{1\over g} {\partial \over\partial y}\) be a rational vector field, where \(f,g\) are homogeneous polynomials of degree \(m,n\) respectively, such that the set \(\{f, g\}\) is a system of parameters of the maximal ideal \((x,y)\) of \(R\). In this paper, the authors propose a concrete method to construct a \(k\)-subalgebra of \(R\), namely the \(k\)-subalgebra \(A\) of \(R\) which is generated by all elements \(\varphi\) of \(R\) such that \(\delta(\varphi)\in R\). The main results are the following: Theorem 1.10. Let \(\widetilde A\) be the integral closure of A in its quotient field. Then the following conditions are equivalent: (1) A is a finitely generated algebra over \(k\); (2) \(f,g\in\widetilde A\); (3) \(\widetilde A=k[x,y]\). Theorem 3.1. Let \(A_0=k[f^ig^j \mid i\geq 2,\;j\geq 2]\). Assume that \(A\cap k[f,g]\supsetneqq A_0\) and that \(\deg f\nmid\deg g\) and \(\deg g\nmid \deg f\). If \(A\) is finitely generated over \(k\) then \(f^M\in A\) or \(g^N\in A\) for some positive integers \(M,N\geq 2\).