On the Skolem Problem for parametric families of linear recurrences

In this talk I will discuss recent results about the Skolem problem for specialisations of linear recurrences defined over a function field. More precisely, given rational functions $a_1,\ldots, a_k, f_1,\ldots, f_k$ defined over a number field, for all but a set of elements $\alpha$ of bounded height in the algebraic closure of $\Q$, the Skolem problem is solvable for the linear recurrence
$F_n(\alpha)=a_1(\alpha)f_1(\alpha)^n + \cdots + a_k(\alpha)f_k(\alpha)^n, \qquad n\ge 0,$
and the existence of a zero can be effectively decided. Moreover, in the case $k=3$, when we restrict to specialisations $\alpha$ that are roots of unity, we give uniform bounds, depending only on the initial data $a_i,f_i$, $i=1,\ldots,k$, on the largest zero $n\ge 0$ such that $F_n(\alpha)= 0$. If time allows I will also briefly discuss connections to certain gcd problems for linear recurrences.

These are joint works with Philipp Habegger, David Masser and Igor Shparlinski.