Diophantine equations with power sums and universal Hilbert sets
type: Item
instance of: scholarly article
title: Diophantine equations with power sums and universal Hilbert sets
zbMATH Open document ID: 0923.11103
DOI: 10.1016/S0019-3577(98)80001-3
published in: Indagationes Mathematicae. New Series
publication date: 1999-06-01T00:00:00Z
review text: For a ring \(A\subset {\mathbb C}\), let \({\mathcal E}_A\) denote the ring of complex functions \(\alpha :{\mathbb N}\rightarrow{\mathbb C}\) of type \(\alpha(n)=\sum_{i=1}^h c_ia_i^n\), where the \(a_i\), called the roots of \(\alpha\), are distinct elements of \(A\) and \(c_i\in{\mathbb Q}\). Let also \({\mathcal E}_A^+\) denote the subring of \({\mathcal E}_A\) formed by those functions which have only positive roots. The authors prove a number of results related to these rings of functions, in the spirit that if a relation of a special kind holds for infinitely many values \(\alpha(n)\), then there is a general relation involving the functions themselves. The main results are the following: (i) if \(\alpha,\beta\in{\mathcal E}_{\mathbb Z}^+\) and \(\alpha(n)/\beta(n)\) is an integer for infinitely many \(n\), then there exists \(\zeta\in {\mathcal E}_{\mathbb Z}^+\) such that \(\alpha=\beta\cdot\zeta\); (ii) if \(F\in{\mathbb Q}[X,Y]\) is an absolutely irreducible polynomial of degree \(d\) in \(Y\) and \(d'\) in \(X\), if \(\alpha\in{\mathcal E}_{\mathbb Z}^+\) and the equation \(F(\alpha(n),Y)=0\) has infinitely many solutions \((n,y_n)\in{\mathbb N}\times{\mathbb Z},\) then all but finitely many of these solutions satisfy a relation of type \(y_n=g(\pm(cb^n)^{1\over d}\lambda(n))\), where \(c\in{\mathbb Q}^*\), \(b\in{\mathbb N}\), \(\lambda\in{\mathcal E}_{\mathbb Z}^+\) and \(g\in{\mathbb Q}[T]\) is a polynomial of degree \(d'\). Again for \(\alpha\in{\mathcal E}_{\mathbb Z}^+\), the paper contains other similar results concerning the approximations of the values \(\alpha(n)\) by \(d\)-th powers of integers and a necessary and sufficient condition for \(\alpha({\mathbb N})\) to be a universal Hilbert set. As pointed out by the authors, some results can be generalized to rings of functions having roots in a ring \(A\) other than \({\mathbb Z}\) and coefficients \(c_i\) in a field other than \({\mathbb Q}\), provided that there is a unique root having maximu…
reviewed by: Roberto Dvornicich
Mathematics Subject Classification ID: 11J25
Mathematics Subject Classification ID: 12E25
Mathematics Subject Classification ID: 11D99
zbMATH DE Number: 1251194
zbMATH Keywords: recurrence relations
zbMATH Keywords: diophantine approximation
zbMATH Keywords: universal Hilbert sets
zbMATH Keywords: diophantine equations with power sums
author: Pietro Corvaja
author: Umberto Zannier
MaRDI profile type: Publication
cites work: Some Problems Concerning Recurrence Sequences
cites work: Q3331274
cites work: Q3853755
cites work: Q3773943
cites work: Q3947744
cites work: Diophantine approximation
cites work: Diophantine approximations and diophantine equations
cites work: Pure powers in recurrence sequences and some related diophantine equations
cites work: Hilbert irreducibility sequences and nonstandard arithmetic
cites work: Q4205488
cites work: Q3773942
description: scientific article; zbMATH DE number 1251194