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Linear forms in \(\mathfrak p\)-adic roots of unity. (English) Zbl 0893.11015
Let \(p\) be a rational prime and let \(\mathbb C_p\) denote the completion of the algebraic closure of \(\mathbb Q_p\). This paper proves the following theorem: For every integer \(n\geq1\) and every family of \(n\) elements \(a_1,\dots,a_n\in\mathbb C_p\), there exists a constant \(c>0\) such that, for any roots of unity \(\zeta_1,\dots,\zeta_n\in\mathbb C_p\), either \(\sum a_i\zeta_i=0\) or \(| \sum a_i\zeta_i| \geq c\). Analogous problems for formal power series fields and for global fields are also proved and discussed, respectively.
This theorem was partly motivated by a question of A. L. Smirnov [St. Petersbg. Math. J. 4, 357-375 (1993); translation from Algebra Anal. 4, 186-209 (1992; Zbl 0814.11056)].
Reviewer: P.Vojta (Berkeley)

11D88 \(p\)-adic and power series fields
11D75 Diophantine inequalities
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