Some results on the Flynn–Poonen–Schaefer conjecture

Abstract

AbstractFor $c \in \mathbb {Q}$ , consider the quadratic polynomial map $\varphi _c(z)=z^2-c$ . Flynn, Poonen, and Schaefer conjectured in 1997 that no rational cycle of $\varphi _c$ 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 $\varphi _c$ 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 $\varphi _c$ in terms of the number s of distinct prime factors of the denominator of c. Finally, we show that the Flynn–Poonen–Schaefer conjecture holds for $\varphi _c$ if $s \le 2$ , i.e., if the denominator of c has at most two distinct prime factors.