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Dynamical uniform boundedness for unicritical polynomials
Abstract
We prove the dynamical uniform boundedness conjecture for unicritical polynomials zd + c over every number field K when d ≥ 4, and over every number field K not containing ℚ(√(−3)) when d = 3. Furthermore, our unconditional bounds on perodic points are effectively computable. In the remaining cases, we show that the dynamical uniform boundedness conjecture is implied by the boundedness of Mordell-Weil ranks over K for a specific family of elliptic curves when d = 3 and for Jacobians of dynatomic curves when d = 2.
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BibTeX
@misc{zhang2026dynamical,
title = {{Dynamical uniform boundedness for unicritical polynomials}},
author = {Robin Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.13431},
eprint = {2609.13431},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13431},
}
RIS
TY - UNPB
TI - Dynamical uniform boundedness for unicritical polynomials
AU - Robin Zhang
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2609.13431
UR - https://arxiv.org/abs/2609.13431
AB - We prove the dynamical uniform boundedness conjecture for unicritical polynomials z^d + c over every number field K when d ≥ 4, and over every number field K not containing ℚ(√(−3)) when d = 3. Furthermore, our unconditional bounds on perodic points are effectively computable. In the remaining cases, we show that the dynamical uniform boundedness conjecture is implied by the boundedness of Mordell-Weil ranks over K for a specific family of elliptic curves when d = 3 and for Jacobians of dynatomic curves when d = 2.
ER -