Poonen's conjecture and Ramsey numbers

Shalom Eliahou (1) , Youssef Fares (2)

1 LMPA - Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville
2 LAMFA - Laboratoire Amiénois de Mathématique Fondamentale et Appliquée - UMR CNRS 7352

For c is an element of Q*, let phi(c) : Q -> Q denote the quadratic map phi(c)(X) = X-2 + c. How large can the period of a rational periodic point of phi(c) be? Poonen conjectured that it cannot exceed 3. Here, we tackle this conjecture by graph-theoretical means with the Ramsey numbers R-k(3). We show that, for any c is an element of Q* whose denominator admits at most k distinct prime factors, the map phi(c) admits at most 2R(k)(3) - 2 periodic points. As an application, we prove that Poonen's conjecture holds for all c is an element of Q* whose denominator is a power of 2. (C) 2015 Elsevier B.V. All rights reserved.

Mots clés: Rational point; Periodic point; Quadratic polynomial; Stability number; Triangle-free graphs

Domaines: Mathématiques [math]

Shalom Eliahou, Youssef Fares. Poonen's conjecture and Ramsey numbers. Discrete Applied Mathematics, 209, pp.102-106, 2016, 9th International Colloquium on Graph Theory and Combinatorics, 2014, Grenoble, ⟨10.1016/j.dam.2015.07.038⟩. ⟨hal-03622019⟩