Simultaneously preperiodic points for a family of polynomials in positive characteristic

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 f_λ(x):=xᵈ+λ (parameterized by λ), given two starting points a and b in C, if there exist infinitely many λ such that both a and b are preperiodic under the action of f_λ, 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 ; (ii) d is a power of p; and (iii) not both a and b live in , 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∈, both a and b are preperiodic under the action of f_λ, while in case~(ii), one obtains that also whenever a-b∈, then for each parameter λ, we have that a is preperiodic under the action of f_λ if and only if b is preperiodic under the action of f_λ.