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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)

##### MSC:
 11D88 $$p$$-adic and power series fields 11D75 Diophantine inequalities
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