Mathematische Zeitschrift

, Volume 263, Issue 2, pp 473–479 | Cite as

Quasi-locally finite polynomial endomorphisms



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.


Polynomial automorphisms Linear recurrent sequences 


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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of La RochelleLa RochelleFrance

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