Diophantine equations with power sums and universal Hilbert sets
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 maximum modulus. The proofs use essentially a quantitative version (proved by Schlickewei) of the subspace theorem [\textit{W. M. Schmidt}, Diophantine approximations and diophantine equations, Lect. Notes Math. 1467 (Springer 1991; Zbl 0754.11020), Theorem 1E, p. 178].
scientific article; zbMATH DE number 1251194
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