The 𝐴𝐵𝐶-Conjecture implies uniform bounds on dynamical Zsigmondy sets

Abstract

We prove that the a b c abc -Conjecture implies upper bounds on Zsigmondy sets that are uniform over families of unicritical polynomials over number fields. As an application, we use the a b c abc -Conjecture to prove that there exist uniform bounds on the index of the associated arboreal Galois representations.