an:05885506
Zbl 1246.37102
Sookdeo, Vijay A.
Integer points in backward orbits
EN
J. Number Theory 131, No. 7, 1229-1239 (2011).
00277985
2011
j
37P35 11S82
arithmetic dynamics; backward orbits; relative \(S\)-integrality; Galois orbits; Galois action on pre-images; dynamic Lehmer's conjecture
Summary: A theorem of J. Silverman states that a forward orbit of a rational map \(\varphi (z)\) on \(\mathbb P^1(K)\) contains finitely many \(S\)-integers in the number field \(K\) when (\(\varphi \circ\varphi )(z)\) 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 \(\varphi (z)=z^d\), and consequently Chebyshev polynomials, by uniformly bounding the number of Galois orbits for \(z^n - \beta \) when \(\beta \neq 0\) is a non-root of unity. In general, our conjecture is true provided that the number of Galois orbits for \(\varphi ^n(z) - \beta \) is bounded independently of \(n\).