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Some results on the Flynn-Poonen-Schaefer Conjecture

Abstract

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.
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Dates and versions

hal-02073665 , version 1 (20-03-2019)

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