Polynomial decomposition and lacunarity

Abstract:

The possible ways of writing a polynomial as a composition of lower degree polynomials were studied by many authors, starting with J.F. Ritt in the 1920's. There are applications to several areas of mathematics. On the other hand, there are many Diophantine problems involving linearly recurrent sequences of polynomials. For a sequence of polynomials \$(G_n(x))_{n=0}^{\infty}\$ in \$\mathbb{C}[x]\$ satisfying a linear recurrence relation of order \$d\geq 2\$:

$$

G_{n+d}(x)=A_{d-1}(x)G_{n+d-1}(x)+\cdots+A_0(x)G_n(x), \ n\in\mathbb{N},

$$

determined by \$A_0,A_1,\ldots,A_{d-1}, G_0,G_1,\ldots,G_{d-1}\in \mathbb{C}[x]\$, one may ask about the properties of \$g(x), h(x)\in \mathbb{C}[x]\$ such that \$G_n(x)=g(h(x))\$. In this talk I will present some results on this topic that come from a joint work with Clemens Fuchs and Christina Karolus from University of Salzburg (Austria). Our work was inspired by Zannier's results about lacunary polynomials. I will further discuss some Diophantine applications of these results.