English
 
Help Privacy Policy Disclaimer
  Advanced SearchBrowse

Item

ITEM ACTIONSEXPORT

Released

Journal Article

Algebraic curves Aol(x) - U(y) = 0 and arithmetic of orbits of rational functions

MPS-Authors
/persons/resource/persons235942

Pakovich,  Fedor
Max Planck Institute for Mathematics, Max Planck Society;

External Resource
Fulltext (restricted access)
There are currently no full texts shared for your IP range.
Fulltext (public)

1801.01985.pdf
(Preprint), 286KB

Supplementary Material (public)
There is no public supplementary material available
Citation

Pakovich, F. (2020). Algebraic curves Aol(x) - U(y) = 0 and arithmetic of orbits of rational functions. Moscow Mathematical Journal, 20(1), 153-183. doi:10.17323/1609-4514-2020-20-1-153-183.


Cite as: https://hdl.handle.net/21.11116/0000-0006-0C2D-C
Abstract
We give a description of pairs of complex rational functions $A$ and $U$ of degree at least two such that for every $d\geq 1$ the algebraic curve $A^{\circ d}(x)-U(y)=0$ has a factor of genus zero or one. In particular, we show that if $A$ is not a `generalized Latt\`es map', then this condition is satisfied if and only if there exists a rational function $V$ such that $U\circ V=A^{\circ l}$ for some $l\geq 1.$ We also prove a version of the dynamical Mordell-Lang conjecture, concerning intersections of orbits of points from $\mathbb P^1(K)$ under iterates of $A$ with the value set $U(\mathbb P^1(K))$, where $A$ and $U$ are rational functions defined over a number field $K.$