Diophantine equations with power sums and universal Hilbert sets.(English)Zbl 0923.11103

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 [W. M. Schmidt, Diophantine approximations and diophantine equations, Lect. Notes Math. 1467 (Springer 1991; Zbl 0754.11020), Theorem 1E, p. 178].

MSC:

 11J25 Diophantine inequalities 12E25 Hilbertian fields; Hilbert’s irreducibility theorem 11D99 Diophantine equations

Zbl 0754.11020
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References:

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