2024/12/12 by Dragos Ghioca · 3 citations
Mathematics · #Algebraic Geometry and Number Theory #Mathematical Dynamics and Fractals #Advanced Differential Equations and Dynamical Systems
paper · doi:10.4153/s0008414x24000841
Abstract In the goundbreaking paper [BD11] (which opened a wide avenue of research regarding unlikely intersections in arithmetic dynamics), Baker and DeMarco prove that for the family of polynomials fλ (x):=xd+λ (parameterized by λ ∈ \mathbb C ), given two starting points a and b in \mathbb C , if there exist infinitely many λ ∈ \mathbb C such that both a and b are preperiodic under the action of fλ , then ad=bd . In this paper, we study the same question, this time working in a field of characteristic p>0 . The answer in positive characteristic is more nuanced, as there are three distinct cases: (i) both starting points a and b live in \mathbb Fp ; (ii) d is a power of p ; and (iii) not both a and b live in \mathbb Fp , while d is not a power of p . Only in case (iii), one derives the same conclusion as in characteristic 0 (i.e., that ad=bd ). In case (i), one has that for each λ ∈ \mathbb Fp , both a and b are preperiodic under the action of fλ , while in case (ii), one obtains that also whenever a-b∈ \mathbb Fp , then for each parameter λ , we have that a is preperiodic under the action of fλ if and only if b is preperiodic under the action of fλ .