Some results on the Flynn-Poonen-Schaefer Conjecture

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 ∈ Q, consider the quadratic polynomial map ϕ_c (x) = x^2 − c. Flynn, Poonen and Schaefer conjectured in 1997 that no rational cycle of ϕ_c under iteration has length more than 3. Here we discuss this conjecture using arithmetic and combinatorial means, leading to three main results. First, we show that if ϕ_c admits a rational cycle of length n ≥ 3, then the denominator of c must be divisible by 16. We then provide an upper bound on the number of periodic rational points of ϕ_c in terms of the number of distinct prime factors of the denominator of c. Finally, we show that the Flynn-Poonen-Schaefer conjecture holds for ϕ_c if that denominator has at most two distinct prime factors.

Domaines: Combinatoire [math.CO]; Théorie des nombres [math.NT]

Shalom Eliahou, Youssef Fares. Some results on the Flynn-Poonen-Schaefer Conjecture. 2019. ⟨hal-02073665⟩