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Simultaneously preperiodic points and colliding orbits for families of polynomials in characteristic p
Abstract
Let k=𝔽̅p and let f∈ k[X] be a polynomial of degree at least $2$ such that f−f(0) is non-additive. For α,β in an algebraically closed extension L of k, we prove that there are infinitely many λ∈ L for which both points are preperiodic under f(X)+λ if and only if α,β∈ k or f(β)−f(α)∈ Ef. Here Ef={c∈ Gf:(f−f(0))n(c)∈ Gf for every n≥0}, where Gf={c∈ k : f(X+c)−f(X)∈ k}. We also provide a necessary and sufficient criterion for the corresponding colliding orbits set to be infinite.
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BibTeX
@misc{lee2026simultaneously,
title = {{Simultaneously preperiodic points and colliding orbits for families of polynomials in characteristic $p$}},
author = {Jungin Lee and GyeongHyeon Nam},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17650},
eprint = {2610.17650},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17650},
}
RIS
TY - UNPB
TI - Simultaneously preperiodic points and colliding orbits for families of polynomials in characteristic p
AU - Jungin Lee
AU - GyeongHyeon Nam
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.17650
UR - https://arxiv.org/abs/2610.17650
AB - Let k=𝔽̅_p and let f∈ k[X] be a polynomial of degree at least $2$ such that f−f(0) is non-additive. For α,β in an algebraically closed extension L of k, we prove that there are infinitely many λ∈ L for which both points are preperiodic under f(X)+λ if and only if α,β∈ k or f(β)−f(α)∈ E_f. Here E_f={c∈ G_f:(f−f(0))^n(c)∈ G_f for every n≥0}, where G_f={c∈ k : f(X+c)−f(X)∈ k}. We also provide a necessary and sufficient criterion for the corresponding colliding orbits set to be infinite.
ER -