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Quasi-locally finite polynomial endomorphisms

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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematische Zeitschrift Springer Journals

Quasi-locally finite polynomial endomorphisms

Mathematische Zeitschrift , Volume 263 (2) – Oct 25, 2008

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References (18)

Publisher
Springer Journals
Copyright
Copyright © 2008 by Springer-Verlag
Subject
Mathematics; Mathematics, general
ISSN
0025-5874
eISSN
1432-1823
DOI
10.1007/s00209-008-0440-4
Publisher site
See Article on Publisher Site

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.

Journal

Mathematische ZeitschriftSpringer Journals

Published: Oct 25, 2008

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