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Collision Parameters in Higher-Dimensional Families of Polynomials
Abstract
Let d≥ 3 and consider the generic family of monic normalized polynomials ft(z)=zd+t1zd−2+t2zd−3+⋯+td−2z+td−1, where t=(t1,…,td−1). Given pairwise distinct starting points α1,…,αd∈ℚ̅ and a target point β∈ℚ̅, we show that the set of parameters λ∈𝔸d−1(ℚ̅) for which β belongs to the forward orbit of every αi under fλ is not Zariski dense in 𝔸d−1.
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BibTeX
@misc{ghioca2026collision,
title = {{Collision Parameters in Higher-Dimensional Families of Polynomials}},
author = {Dragos Ghioca and Negin Shadgar},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05785},
eprint = {2611.05785},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05785},
}
RIS
TY - UNPB
TI - Collision Parameters in Higher-Dimensional Families of Polynomials
AU - Dragos Ghioca
AU - Negin Shadgar
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.05785
UR - https://arxiv.org/abs/2611.05785
AB - Let d≥ 3 and consider the generic family of monic normalized polynomials f_t(z)=z^d+t_1z^d−2+t_2z^d−3+⋯+t_d−2z+t_d−1, where t=(t_1,…,t_d−1). Given pairwise distinct starting points α_1,…,α_d∈ℚ̅ and a target point β∈ℚ̅, we show that the set of parameters λ∈𝔸^d−1(ℚ̅) for which β belongs to the forward orbit of every α_i under f_λ is not Zariski dense in 𝔸^d−1.
ER -