The Newton polygon of a recurrence sequence of polynomials and its role in TQFT

Abstract: The paper contains a combinatorial theorem (the sequence of Newton polygons of a reccurent sequence of polynomials is quasi-linear) and two applications of it in classical and quantum topology, namely in the behavior of the $A$-polynomial and a fixed quantum invariant (such as the Jones polynomial) under filling. Our combinatorial theorem, which complements results of Calegari-Walker \cite{CW} and the author \cite{Ga4}, occupies the bulk of the paper and its proof requires the Lech-Mahler-Skolem theorem of $p$-adic analytic number theory combined with basic principles in polyhedral and tropical geometry.