Primitive divisors in arithmetic dynamics
By Patrick Ingram, Joseph H. Silverman. Mathematical Proceedings of the Cambridge Philosophical Society, 2007
52 citations recorded by the source corpus.
Abstract Let ϕ(z) ∈ (z) be a rational function of degree d ≥ 2 with ϕ(0) = 0 and such that ϕ does not vanish to order d at 0. Let α ∈ have infinite orbit under iteration of ϕ and write ϕn(α) = An/Bn as a fraction in lowest terms. We prove that for all but finitely many n ≥ 0, the numerator An has a primitive divisor, i.e., there is a prime p such that p | An and p ∤ Ai for all i < n. More generally, we prove an analogous result when ϕ is defined over a number field and 0 is a preperiodic point for ϕ.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/5dc79dc1d661c12a018650797589824a5bcd09b8
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Geometry and Topology
- T. Tucker
- Robert L. Benedetto
- Laura Demarco
- Joseph H. Silverman
- Michelle Manes
- Rafe Jones
- Patrick Ingram
- Franccois Berteloot
In Mathematical Physics
- T. Tucker
- Robert L. Benedetto
- Michelle Manes
- Rafe Jones
- Laura Demarco
- Patrick Ingram
- Joseph H. Silverman
- Franccois Berteloot
Citations in the corpus
- Current trends and open problems in arithmetic dynamics (2018, background): “(Ingram–Silverman, 2009, [120]) Let f(x) ∈ Q(x) be a rational function of degree at least 2, let α ∈ Q be an f -wandering point, and let β ∈ Q be an f -preperiodic point.”
- Galois representations from pre-image trees: an arboreal survey (2014, background): “For a sampling of the large and interesting literature on this question, which merits a survey of its own, see [10, 12, 18, 26, 42, 45].”