Decomposable polynomials in linear recurrence sequences

Dijana Kreso (TU Graz)

Frank Adams 1,

In this talk I will present results that come from an ongoing joint work with Clemens Fuchs and Christina Karolus from University of Salzburg (Austria). We study elements of second order linear recurrence sequences \($(G_n)_{n= 0}^{\infty}\) of complex polynomials which are decomposable, i.e. representable as \(G_n=g\circ h\) for some \(g, h\in \mathbb{C}[x]\) satisfying \(\deg g,\deg h>1\). Under certain assumptions, and provided that \(h\) is not of particular type, we show that \(\deg g\) may be bounded by a constant independent of \(n\). Our result is similar in flavor to a result of Zannier from 2007 who showed that if \(f\) is a polynomial with \(\ell\) non-constant terms, \(f(x)=g(h(x))\), and \(h(x)\) is not of type \(ax^k+b\) with \(a\neq 0\), then \(\deg g\leq 2\ell(\ell-1)\). The possible ways of writing a polynomial as a composition of lower degree polynomials were studied by several authors, starting with the American mathematician Ritt in the 1920's. Results in this area of mathematics have applications to various other areas, e.g. number theory, complex analysis, etc. In my talk I will present some Diophantine applications.
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