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On Diophantine approximations of the solutions of q-functional equations

Published online by Cambridge University Press:  12 July 2007

Tapani Matala-aho
Affiliation:
Matemaattisten tieteiden laitos, Linnanmaa, PL 3000, 90014 Oulun Yliopisto, Finland (tma@cc.oulu.fi)
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Abstract

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Given a sequence of linear forms in m ≥ 2 complex or p-adic numbers α1, …,αm ∈ Kv with appropriate growth conditions, Nesterenko proved a lower bound for the dimension d of the vector space Kα1 + ··· + Kαm over K, when K = Q and v is the infinite place. We shall generalize Nesterenko's dimension estimate over number fields K with appropriate places v, if the lower bound condition for |Rn| is replaced by the determinant condition. For the q-series approximations also a linear independence measure is given for the d linearly independent numbers. As an application we prove that the initial values F(t), F(qt), …, F(qm−1t) of the linear homogeneous q-functional equation where N = N(q, t), Pi = Pi(q, t) ∈ K[q, t] (i = 1, …, m), generate a vector space of dimension d ≥ 2 over K under some conditions for the coefficient polynomials, the solution F(t) and t, q ∈ K*.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2002