2025/08/08 by Asgarli, Shamil, Ghioca, Dragos · 1 citation
#11T06 #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT) #Primary 37P05 #Secondary 37P30
paper · doi:10.48550/arxiv.2508.06279
Let L be a field of positive characteristic p with a fixed algebraic closure L, and let α1,α2,β∈ L. For an integer d≥ 2, we consider the family of polynomials fλ(z) := zd+λ, parameterized by λ∈L. Define C(α1,α2;β) to be the set of all λ∈L for which there exist m,n∈ℕ such that fλm(α1)=fλn(α2)=β. In other words, C(α1,α2;β) consists of all λ∈L with the property that the orbit of α1 collides with the orbit of α2 under the same polynomial fλ precisely at the point β. Assuming α1,α2,β are not all contained in a finite subfield of L, we provide explicit necessary and sufficient conditions under which C(α1,α2;β) is infinite. We also discuss the remaining case where α1,α2,β∈ \mathbb Fp and provide ample computational data that suggest a somewhat surprising conjecture. Our problem fits into a long series of questions in the area of unlikely intersections in arithmetic dynamics, which have been primarily studied over fields of characteristic 0. Working in characteristic p adds significant difficulties, but also reveals the subtlety of our problem, especially when some of the points lie in a finite field or when d is a power of p.