Approximation of values of algebraic elements over the ring of power sums

Abstract

Let $\mathbb{Q}{\mathcal{E}}_{\mathbb{Z}}$ be the set of power sums whose characteristic roots belong to $\mathbb{Z}$ and whose coefficients belong to $\mathbb{Q}$, i.e. $G:\mathbb{N}\to \mathbb{Q}$ satisfies \begin{equation*} G(n) = G_n = b_1 c_1^n + \cdots + b_h c_h^n \end{equation*} with $c_1,\dots ,c_h\in \mathbb{Z}$ and $b_1,\dots ,b_h\in \mathbb{Q}$. Furthermore, let $f\in \mathbb{Q}[x,y]$ be absolutely irreducible and $\alpha :\mathbb{N}\to \overline{\mathbb{Q}}$ be a solution $y$ of $f(G_n,y)=0$, i.e. $f(G_n,\alpha (n))=0$ identically in $n$. Then we will prove under suitable assumptions a lower bound, valid for all but finitely many positive integers $n$, for the approximation error if $\alpha (n)$ is approximated by rational numbers with bounded denominator. After that we will also consider the case that $\alpha$ is a solution of \begin{equation*} f(G_n^{(0)}, \ldots, G_n^{(d)},y) = 0, \end{equation*} i.e. defined by using more than one power sum and a polynomial $f$ satisfying some suitable conditions. This extends results of Bugeaud, Corvaja, Luca, Scremin and Zannier.

19 pages

References in corpus (1)

• Integral zeros of a polynomial with linear recurrences as coefficients

Cited by in corpus (1)

• Some arithmetical properties of convergents to algebraic numbers