On the growth of linear recurrences in function fields

TL;DR

This paper establishes a function field analogue of a classical growth result for linear recurrence sequences, showing that for large n, the sequence's magnitude closely approximates the dominant root's exponential growth.

Contribution

It proves a new growth inequality for linear recurrences over function fields, extending classical number field results to a broader algebraic setting.

Findings

For large n, |G_n| approximates the dominant root's exponential growth.

The inequality |G_n| ≥ (max |α_j|)^{n(1-ε)} holds under certain conditions.

The result bridges classical recurrence growth results with function field analogues.

Abstract

Let $ (G_n)_{n=0}^{\infty} $ be a non-degenerate linear recurrence sequence with power sum representation $ G_n = a_1(n) \alpha_1^n + \cdots + a_t(n) \alpha_t^n $. In this paper we will prove a function field analogue of the well known result that in the number field case, under some non-restrictive conditions, for $ n $ large enough the inequality $ \vert G_n\vert \geq \left( \max_{j=1,\ldots,t} \vert \alpha_j\vert \right)^{n(1-\varepsilon)} $ holds true.