Some results on the Flynn–Poonen–Schaefer conjecture
Canadian mathematical bulletin, Tome 65 (2022) no. 3, pp. 598-611
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For $c \in \mathbb {Q}$, 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.
Mots-clés :
Rational quadratic polynomial, polynomial iteration, discrete dynamical system, periodic points
Eliahou, Shalom; Fares, Youssef. Some results on the Flynn–Poonen–Schaefer conjecture. Canadian mathematical bulletin, Tome 65 (2022) no. 3, pp. 598-611. doi: 10.4153/S0008439521000588
@article{10_4153_S0008439521000588,
author = {Eliahou, Shalom and Fares, Youssef},
title = {Some results on the {Flynn{\textendash}Poonen{\textendash}Schaefer} conjecture},
journal = {Canadian mathematical bulletin},
pages = {598--611},
year = {2022},
volume = {65},
number = {3},
doi = {10.4153/S0008439521000588},
url = {http://geodesic.mathdoc.fr/articles/10.4153/S0008439521000588/}
}
TY - JOUR AU - Eliahou, Shalom AU - Fares, Youssef TI - Some results on the Flynn–Poonen–Schaefer conjecture JO - Canadian mathematical bulletin PY - 2022 SP - 598 EP - 611 VL - 65 IS - 3 UR - http://geodesic.mathdoc.fr/articles/10.4153/S0008439521000588/ DO - 10.4153/S0008439521000588 ID - 10_4153_S0008439521000588 ER -
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