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The average number of rational preperiodic points of polynomials over ℚ
Abstract
Let Md(X) denote the average number of rational preperiodic points among degree-d polynomials over ℚ with vanishing zd−1 coefficient, constant term $1$, and height at most X. We prove that, for every integer d≥2, Md(X) ∼ γd X−1 for an explicit constant γd>0. For d≥4, the error term is Od,ε(X−2+ε) for every ε>0; for d=3 the error term is O(X−3/2); and for d=2 the error term is O(X−3/2log X).
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BibTeX
@misc{lee2026average,
title = {{The average number of rational preperiodic points of polynomials over $\mathbbQ$}},
author = {Jungin Lee},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.29383},
eprint = {2608.29383},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29383},
}
RIS
TY - UNPB
TI - The average number of rational preperiodic points of polynomials over ℚ
AU - Jungin Lee
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2608.29383
UR - https://arxiv.org/abs/2608.29383
AB - Let M_d(X) denote the average number of rational preperiodic points among degree-d polynomials over ℚ with vanishing z^d−1 coefficient, constant term $1$, and height at most X. We prove that, for every integer d≥2, M_d(X) ∼ γ_d X^−1 for an explicit constant γ_d>0. For d≥4, the error term is O_d,ε(X^−2+ε) for every ε>0; for d=3 the error term is O(X^−3/2); and for d=2 the error term is O(X^−3/2log X).
ER -