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Rank-two recurrence results for polynomials and questions of dynamical Mordell--Lang type
Abstract
Let f,g∈ℂ[z]∖ℂ and c∈ℂ[z]. Suppose that deg(c)=1 if deg(f)=deg(g)=1. Using the theory of Presburger arithmetic, we prove that the rank-two recurrence set Sf,g,c2:={(m,n)∈ℤ≥02: ∃λ∈ℂ, f∘ m(λ)=g∘ n(λ)=c(λ)} is semi-linear. This is a generalization of a theorem of Yang and Zhong for the case m=n. We also obtain partial results on recurrence sets for rational maps. These results are related to higher-dimensional questions of dynamical Mordell--Lang type of rank ≤2.
Related
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- Local height arguments toward the dynamical Mordell-Lang conjecture
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BibTeX
@misc{zhang2026ranktwo,
title = {{Rank-two recurrence results for polynomials and questions of dynamical Mordell--Lang type}},
author = {Geng-Rui Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.27058},
eprint = {2605.27058},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.27058},
}
RIS
TY - UNPB
TI - Rank-two recurrence results for polynomials and questions of dynamical Mordell--Lang type
AU - Geng-Rui Zhang
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2605.27058
UR - https://arxiv.org/abs/2605.27058
AB - Let f,g∈ℂ[z]∖ℂ and c∈ℂ[z]. Suppose that deg(c)=1 if deg(f)=deg(g)=1. Using the theory of Presburger arithmetic, we prove that the rank-two recurrence set S_f,g,c^2:={(m,n)∈ℤ_≥0^2: ∃λ∈ℂ, f^∘ m(λ)=g^∘ n(λ)=c(λ)} is semi-linear. This is a generalization of a theorem of Yang and Zhong for the case m=n. We also obtain partial results on recurrence sets for rational maps. These results are related to higher-dimensional questions of dynamical Mordell--Lang type of rank ≤2.
ER -