The field of iterates of a rational function
Francesco Veneziano1 ; Solomon Vishkautsan2
1 Department of mathematics, University of Genova, Via Dodecaneso 35, 16146 Genova, Italy
2 Department of Computer Science, Tel-Hai Academic College, Upper Galilee 9977, Qiryat Shemona 1220800, Israel
Journal de théorie des nombres de Bordeaux, Tome 37 (2025) no. 2, pp. 691-709

We study how the field of definition of a rational function changes under iteration. We provide a complete classification of polynomials with the property that the field of definition of one of their iterates drops in degree (over a given base field). We show with families of examples that this characterization does not hold for rational functions. Finally, we also classify fractional linear transformations with this property.

Nous étudions comment le corps de définition d’une fonction rationnelle évolue sous itération. Nous fournissons une classification complète des polynômes ayant la propriété que le corps de définition de l’un de leurs itérés diminue en degré (par rapport à un corps de base donné). Nous montrons, à l’aide de familles d’exemples, que cette caractérisation ne s’applique pas aux fonctions rationnelles. Enfin, nous classifions également les transformations linéaires fractionnaires ayant cette propriété.

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DOI : 10.5802/jtnb.1337
Classification : 37P05, 11C08, 37P15
Keywords: field of definition, iterates of rational functions, arithmetic dynamics

Francesco Veneziano 1 ; Solomon Vishkautsan 2

1 Department of mathematics, University of Genova, Via Dodecaneso 35, 16146 Genova, Italy
2 Department of Computer Science, Tel-Hai Academic College, Upper Galilee 9977, Qiryat Shemona 1220800, Israel
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Francesco Veneziano; Solomon Vishkautsan. The field of iterates of a rational function. Journal de théorie des nombres de Bordeaux, Tome 37 (2025) no. 2, pp. 691-709. doi: 10.5802/jtnb.1337

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