A survey on the hypertranscendence of the solutions of the Schröder's, Böttcher's and Abel's equations

Gwladys Fernandes (1)

1 LMV - Laboratoire de Mathématiques de Versailles

In 1994, P.-G. Becker and W. Bergweiler listed all the differentially algebraic solutions of three famous functional equations: the Schröder's, Böttcher's and Abel's equations. The proof of this theorem combines various domains of mathematics. This goes from the theory of iteration, which gave birth to these equations, to the algebro-differential notion of coherent families developed by M. Boshernitzan and L. A. Rubel. This survey is an excursion into the history of these equations, in order to enlighten the different pieces of mathematics they bring together and how these parts fit into the result of P.-G. Becker and W. Bergweiler.

Domaines: Théorie des nombres [math.NT]

Gwladys Fernandes. A survey on the hypertranscendence of the solutions of the Schröder's, Böttcher's and Abel's equations. TRANS 2019: Transcendence in Algebra, Combinatorics, Geometry and Number Theory, Springer Proceedings in Mathematics & Statistics, 373, Springer International Publishing, pp.91-125, 2021, Springer Proceedings in Mathematics & Statistics, ⟨10.1007/978-3-030-84304-5_4⟩. ⟨hal-03090954⟩