Rational periodic points of xd + c and Fermat–Catalan equations

Abstract

In this paper, we study rational periodic points of polynomial [Formula: see text] over the field of rational numbers, where [Formula: see text] is an integer greater than two. For period two, we describe periodic points for degrees [Formula: see text]. We also demonstrate the nonexistence of rational periodic points of exact period two for [Formula: see text] such that [Formula: see text] and [Formula: see text] has a prime factor greater than three. Moreover, assuming the [Formula: see text]-conjecture, we prove that [Formula: see text] has no rational periodic point of exact period greater than one for sufficiently large integer [Formula: see text] and [Formula: see text].

The abcd conjecture, uniform boundedness, and dynamical systems

“…For periodic points of unicritical polynomials with coefficients in O K , Panraksa observed in an unpublished note [53] that Theorem 3.12 implies a uniform bound on possible periods. where ϵ p (ϕ) is the greatest exponent of p dividing the resultant of the PGL 2 -conjugate of ϕ with the best reduction at p (cf.…”

The Uniform Boundedness and Dynamical Lang Conjectures for polynomials

“…Aside from the special case of Lattès maps and the unicritical maps studied in [20,24], progress on Conjecture 1.1 has only been obtained by imposing strong local conditions on the dynamics of f , as in [3,8,16]. We will forfeit all such local hypotheses, at the expense of assuming a standard conjecture in arithmetic geometry, in order to prove Conjecture 1.1 for the K-rational preperiodic points of polynomials.…”