Quasi-locally finite polynomial endomorphisms (Q841109)

A recent joint paper by the author of the paper under review [\textit{J.-P. Furter} and \textit{S. Maubach}, J. Pure Appl. Algebra 211, No. 2, 445--458 (2007; Zbl 1127.14054)] introduced locally finite polynomial endomorphisms of the affine space \({\mathbb C}^N\). They were characterized with several equivalent conditions. One of these conditions says that \(F\in\text{End}({\mathbb C}^N)\) is locally finite if and only if there is a nonzero polynomial \(p(T)\in {\mathbb C}[T]\) such that the linear operator \(p(F)\) is equal to 0 in \({\mathbb C}^N\). In the present paper the author introduces the less restrictive notion of a quasi-locally finite (QLF) endomorphism. Now \(F\in\text{End}({\mathbb C}^N)\) is QLF if \(p(F)=0\) for a polynomial \(p(T)\) with coefficients of the field of rational \(F\)-invariants \({\mathbb C}(X)^F\). The main result is that \(F\) is QLF if and only if the sequence \(F^n(a)\), \(n=1,2,\ldots\), satisfies a linear recurrence for any \(a\in{\mathbb C}^N\). Several nice properties of QLF endomorphisms are established. For example they satisfy the Jacobian conjecture.