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Algebraic independence of \(p\)-adic numbers. (English. Russian original) Zbl 1145.11055

Izv. Math. 72, No. 3, 565-579 (2008); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 72, No. 3, 159-174 (2008).
For a prime number \(p\), let \(\mathbb C_p\) be the completion of the algebraic closure of the field of \(p\)-adic numbers. Suppose that \(a_1\), …, \(a_u\) and \(b_1\), …, \(b_v\) in \(\mathbb C_p\) are such that the inequalities
\[ | k_1a_1+\cdots+ k_ua_u| _p \geq e^{-C(\log K)^\kappa},\qquad | \ell_1b_1+\cdots+ \ell_v b_v| _p \geq e^{-C(\log L)^\kappa}, \]
hold for all large \(K\) and \(L\) and all integers \(k_1,\dots k_u\), \(\ell_1,\dots\ell_v\) with \(0<\max | k_i| \leq K\) and \(0<\max | \ell_j| \leq L\) (for some fixed positive real numbers \(C\) and \(\kappa\) depanding on the \(a_i\)’s and \(b_j\)’s). Then the author proves the following result.
Theorem. Let that \(a_1,\dots,a_u\) and \(b_1,\dots,b_v\) in \(\mathbb C_p\) as above, and assume moreover that \(| a_i| _p <p^{-1/(p-1)}\) and \(| b_j| _p\leq 1\) for all \(i\) and \(j\), then
\[ \text{trdeg}_{\mathbb Q}\mathbb Q\bigl(a_1, \dots, a_u, b_1, \dots, b_v,e^{a_1b_1}, \dots, e^{a_ub_v} \bigr) \geq {uv\over u+v}. \]

MSC:

11J85 Algebraic independence; Gel’fond’s method
11J81 Transcendence (general theory)
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