Effective bounds for the zeros of linear recurrences in function fields

Fuchs, Clemens; Pethő, Attila

Journal de théorie des nombres de Bordeaux, Tome 17 (2005) no. 3, pp. 749-766.

Dans cet article, on utilise la généralisation de l’inégalité de Mason (due à Brownawell et Masser [8]) afin d’exhiber des bornes supérieures effectives pour les zéros d’une suite linéaire récurrente définie sur un corps de fonctions à une variable.

De plus, on étudie de problèmes similairs dans ce contexte, comme l’équation $G_n\left(x\right)=G_m\left(P\left(x\right)\right),(m,n)\in {\mathbb{N}}^2$, où $\left(G_n\left(x\right)\right)$ est une suite récurrente de polynômes et $P\left(x\right)$ un polynôme fixé. Ce problème a été étudié auparavant dans [14,15,16,17,32].

In this paper, we use the generalisation of Mason’s inequality due to Brownawell and Masser (cf. [8]) to prove effective upper bounds for the zeros of a linear recurring sequence defined over a field of functions in one variable.

Moreover, we study similar problems in this context as the equation $G_n\left(x\right)=G_m\left(P\left(x\right)\right),(m,n)\in {\mathbb{N}}^2$, where $\left(G_n\left(x\right)\right)$ is a linear recurring sequence of polynomials and $P\left(x\right)$ is a fixed polynomial. This problem was studied earlier in [14,15,16,17,32].

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