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A note on complex \(p\)-adic exponential fields. (English) Zbl 1436.11086
The author proves the following result. Let \(n\) be a positive integer, \(\mathbb P\) be the set of all primes, \(\mathbb C_p\) be the completion of \(\mathbb Q^{\mathrm{alg}}_p\) and
\[E_p^n=\{ \overline x\in C_p^n; \mid\overline x\mid_p =\max_{1\leq i\leq n} \mid x_i\mid_p <p^{-\frac 1{p-1}}\}. \]
Let \(V\) be a \(\mathbb Q\)-variety of dimension \(n\) in an affine \(2n\) space. Assume that for infinitely many primes \(p\), \(V\) has a \(\mathbb C_p\)-point of the form \((\overline a_p, \exp_p(\overline a_p))\), then there exist a finite set \(S(V)\subset\mathbb P\) and a finite set of rational tuples \(\overline\alpha_i\), \(i\in I\), (where \(I\) is a finite set) such that for all \(p\in \mathbb P\setminus S(V)\) and for all \(n\)-tuples \(\overline x_p\in E_p^n\) satisfying \((\overline x_p, \exp_p(\overline x_p))\in V(C_p)\), there is a rational linear dependence that holds for the tuple \(\overline x_p\) of the form \(\sum_{j=1}^n \alpha_{i,j} x_{p,j}=0\) for some \(i\in I\).
MSC:
11J85 Algebraic independence; Gel’fond’s method
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