arXiv subjects
Publications and source records attributed to Vijay A. Sookdeo.
For a Lattès map $ϕ:\mathbb P^1 \to \mathbb P^1$ defined over a number field $K$, we prove a conjecture on the integrality of points in the backward orbit of $P\in \mathbb P^1(\overline K)$ under $ϕ$.
We show that the backward orbit conjecture is true for powering map $ϕ(z)=z^d$ over a function field $K$ with a finite field of constants, and when $d$ is relatively prime to the characteristic of $K$.
A theorem of J. Silverman states that a forward orbit of a rational map $ϕ(z)$ on $\mathbb P^1(K)$ contains finitely many $S$-integers in the number field $K$ when $(ϕ\circϕ)(z)$ is not a polynomial. We state an analogous conjecture for the backward orbits using a general $S$-integrality notion based on the Galois conjugates of points. This conjecture is proven for the map $ϕ(z)=z^d$, and consequently Chebyshev polynomials, by uniformly bounding the number of Galois orbits for $z^n-β$ when $β\not =0$ is a non-root of unity. In general, our conjecture is true provided that the number of Galois orbits for $ϕ^n(z)-β$ is bounded independently of $n$.