Observational analysis shows the field of definition varies in iterative rational functions, indicating distinct properties compared to polynomials.
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.
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Veneziano et al. (2025) studied this question.
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