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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.
Mathematische Zeitschrift – Springer Journals
Published: Oct 25, 2008
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