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

TL;DR

This paper establishes lower bounds on the approximation errors of algebraic elements over power sum rings, extending previous results to more complex polynomial relations involving multiple power sums.

Contribution

It generalizes existing approximation bounds to algebraic elements defined by multivariate polynomial relations involving multiple power sums.

Findings

Provides lower bounds for approximation errors for almost all n

Extends previous results to multivariate power sum relations

Applicable to algebraic solutions of complex polynomial equations

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} \rightarrow \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,\ldots,c_h \in \mathbb{Z} $ and $ b_1,\ldots,b_h \in \mathbb{Q} $. Furthermore, let $ f \in \mathbb{Q}[x,y] $ be absolutely irreducible and $ \alpha : \mathbb{N} \rightarrow \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.