Rank-two recurrence results for polynomials and questions of dynamical Mordell--Lang type
Fuente:
arXiv
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2026
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866918524682764288 |
|---|---|
| author | Zhang, Geng-Rui |
| author_facet | Zhang, Geng-Rui |
| contents | Let $f,g\in\mathbb{C}[z]\setminus\mathbb{C}$ and $c\in\mathbb{C}[z]$. Suppose that $\mathrm{deg}(c)=1$ if $\mathrm{deg}(f)=\mathrm{deg}(g)=1$. Using the theory of Presburger arithmetic, we prove that the rank-two recurrence set
\[S_{f,g,c}^2:=\left\lbrace(m,n)\in\mathbb{Z}_{\geq0}^2\colon \existsλ\in\mathbb{C}, f^{\circ m}(λ)=g^{\circ n}(λ)=c(λ)\right\rbrace\] 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 in the case $m=n$. These results are related to higher-dimensional questions of dynamical Mordell--Lang type of rank $\leq2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_27058 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Rank-two recurrence results for polynomials and questions of dynamical Mordell--Lang type Zhang, Geng-Rui Dynamical Systems Algebraic Geometry Logic Number Theory Primary 37P05, Secondary 11U09, 37P30, 37F10 Let $f,g\in\mathbb{C}[z]\setminus\mathbb{C}$ and $c\in\mathbb{C}[z]$. Suppose that $\mathrm{deg}(c)=1$ if $\mathrm{deg}(f)=\mathrm{deg}(g)=1$. Using the theory of Presburger arithmetic, we prove that the rank-two recurrence set \[S_{f,g,c}^2:=\left\lbrace(m,n)\in\mathbb{Z}_{\geq0}^2\colon \existsλ\in\mathbb{C}, f^{\circ m}(λ)=g^{\circ n}(λ)=c(λ)\right\rbrace\] 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 in the case $m=n$. These results are related to higher-dimensional questions of dynamical Mordell--Lang type of rank $\leq2$. |
| title | Rank-two recurrence results for polynomials and questions of dynamical Mordell--Lang type |
| topic | Dynamical Systems Algebraic Geometry Logic Number Theory Primary 37P05, Secondary 11U09, 37P30, 37F10 |
| url | https://arxiv.org/abs/2605.27058 |