Commutative bases of derivations in polynomial and power series rings
Let \(k\) be a commutative ring of characteristic zero and let \(R\) denote either the ring \(k[x_ 1,...,x_ n]\) of polynomials over \(k\) or the ring \(kx_ 1,...,x_ n\) of formal power series over \(k\). The \(R\)-module of all \(k\)-derivations from \(R\) to \(R\) is denoted by \(\text{Der}_ k(R)\). A basis \(\{d_ 1,...,d_ n\}\) of \(\text{Der}_ k(R)\) is said to be commutative if \(d_ id_ j=d_ jd_ i\) for any \(i,j=1,...,n\). The following characterization of a commutative basis of \(\text{Der}_ k(R)\) is given: The set \(\{d_ 1,...,d_ n\}\) of \(k\)-derivations of \(R\) is a commutative bases of \(\text{Der}_ k(R)\) iff there exist elements \(F_ 1,...,F_ n\in R\) such that \(d_ i(F_ j)=\delta_{ij}\), for any \(i,j=1,...,n\), where \(\delta_{ij}\) is the Kronecker symbol. Furthermore, two equivalent versions of the Jacobian conjecture in \(k[x_ 1,...,x_ n]\) are given by means of commutative bases in \(\text{Der}_ k(k[x_ 1,..,x_ n])\).
scientific article; zbMATH DE number 3950671