Armand Noubissie: Diophantine analysis over global fields

Diophantine analysis over global fields

Armand Noubissie, Graz University of Technology

Abstract: The generalized Skolem's Problem asks to determine whether a given rational function defined over a global field which does not have a pole at $0$ has a zero term in its power series expansion. Fuchs and Pethő proved this problem is decidable over a function field but the question is still open over number fields. In 1977, an extension of this problem was made by Loxton and Van der Poorten who conjectured that for any $\epsilon >0$ and ${u_n}$ an integer linear recurrence sequence with dominant (s) roots $>1$ in absolute value, there is an effectively computable constant $C(\epsilon),$ such that if $\vert u_n \vert < (\max_i{ \vert \alpha_i \vert })^{n(1-\epsilon)}$, then $n < C(\epsilon)$. Using results of Schmidt and Evertse, a complete non-effective (qualitative) proof of this conjecture was given by Fuchs and Heintze (2021) and, independently, by Karimov and al. (2023). In this talk, we prove a quantitative version of that result by giving an explicit upper bound for the number of solutions. Moreover, we give a function field analogue on growth of multi-recurrence, answering a question posed by Fuchs and Heintze and generalizing the result of Fuchs and Pethő.

This is based on joint work with Clemens Fuchs.