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The paper contains a combinatorial theorem (the sequence of Newton polygonsof a reccurent sequence of polynomials is quasi-linear) and two applications ofit 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 ofCalegari-Walker \cite{CW} and the author \cite{Ga4}, occupies the bulk of thepaper and its proof requires the Lech-Mahler-Skolem theorem of $p$-adicanalytic number theory combined with basic principles in polyhedral andtropical geometry.

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