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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Ensembles sur lesquels les polynômes
sont déterminés par leur image


Author: Michel Savoyant
Journal: Proc. Amer. Math. Soc. 126 (1998), 1143-1148
MSC (1991): Primary 30C10
MathSciNet review: 1425137
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Abstract: Let $A$ be a non-empty subset of the complex plane $\mathbb{C}$, and $P$, $Q$ two complex polynomials. If $P$ and $Q$ having the same image on $A$ implies $P=Q$, we say that $A$ is a generalized unicity set (for polynomials). We construct in this paper a subset $A$ of $\mathbb{C}$ such that $A$ and $\mathbb{C}\setminus A$ are generalized unicity sets, and we give an example of a generalized unicity set which is open, connected and unbounded.

RÉSUMÉ. Soit $A$ un sous-ensemble non vide du plan complexe $\mathbb{C}$, et $P$, $Q$ deux fonctions polynômes à coefficients complexes. Si l'égalité $P(A)=Q(A)$ entraîne $P=Q$, on dira que $A$ est un ensemble d'unicité généralisée (pour les polynômes). On construit dans cet article un sous-ensemble $A$ de $\mathbb{C}$ tel que $A$ et $\mathbb{C}\setminus A$ sont d'unicité généralisée, et on donne aussi l'exemple d'un ensemble d'unicité généralisée qui est ouvert, connexe et non borné.


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Additional Information

Michel Savoyant
Email: savoyant@math.univ-montp2.fr

DOI: http://dx.doi.org/10.1090/S0002-9939-98-04178-1
PII: S 0002-9939(98)04178-1
Received by editor(s): January 29, 1996
Received by editor(s) in revised form: October 1, 1996
Communicated by: Albert Baernstein II
Article copyright: © Copyright 1998 American Mathematical Society