Some Results On The Flynn-Poonen-Schaefer Conjecture

Abstract

For $c\in \mathbb{Q}$, consider the quadratic polynomial map ${\phi }_c(x)=x^2-c$. Flynn, Poonen and Schaefer conjectured in 1997 that no rational cycle of ${\phi }_c$ 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 ${\phi }_c$ admits a rational cycle of length $n\ge 3$, then the denominator of $c$ must be divisible by $16$. We then provide an upper bound on the number of periodic rational points of ${\phi }_c$ 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 ${\phi }_c$ if that denominator has at most two distinct prime factors.