Quasi-locally finite polynomial endomorphisms

Abstract

If F is a polynomial endomorphism of ${\mathbb {C}^N}$, let ${\mathbb {C} (X)^F}$ denote the field of rational functions ${r \in \mathbb C (x_1,\ldots,x_N)}$ such that ${r \circ F=r}$. We will say that F is quasi-locally finite if there exists a nonzero ${p \in \mathbb C (X)^F[T]}$ such that p(F) = 0. This terminology comes out from the fact that this definition is less restrictive than the one of locally finite endomorphisms made in Furter, Maubach (J Pure Appl Algebra 211(2):445–458, 2007). Indeed, F is called locally finite if there exists a nonzero ${p \in \mathbb C [T]}$ such that p(F) = 0. In the present paper, we show that F is quasi-locally finite if and only if for each ${a \in \mathbb C^N}$ the sequence ${n \mapsto F^n(a)}$ is a linear recurrent sequence. Therefore, this notion is in some sense natural. We also give a few basic results on such endomorphisms. For example: they satisfy the Jacobian conjecture.