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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
##### Keywords:
$$p$$-adic exponential; Ax-Schanuel theorem; ultraproducts
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##### References:
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