Properties of elements of direct products of fields

V. Yu. Matveev

Čebyševskij sbornik, Tome 20 (2019) no. 2, pp. 383-390.

Résumé

The paper describes certain arithmetic properties of values of $F$-series, i.e. of series of the form \begin{equation} \nonumber \sum_{n=0}^\infty a_n \cdot n! \; z^n. \end{equation} Here $a_n\in\mathbb K$, a certain algebraic number field of a finite degree over $\mathbb Q$. The maximum of the absolute values of the conjugates to $a_n$ doesn't exceed $e^{C_1 n}$. Also there exists a sequence of rational integers $d_n = d_{0,n} q^n$, $q\in\mathbb N$, $n=0,1,\ldots$ such that $d_n a_k\in\mathbb Z_{\mathbb K}$, $n=0,1,\ldots$, $k=0,1,\ldots,n$. Meanwhile $d_{0,n}$ is divisible only by primes $p$, $p\leqslant C_2 n$ and \begin{equation} \nonumber ord_p d_{0,n} \leqslant C_3\left(\log_p^n + \frac{n}{p^2}\right). \end{equation} Some general theorem is proved in analogy to Salikhov's theorem for the $E$-functions. It gives conditions of the algebraic independence over $\mathbb C(z)$ of a set of $F$-series, each being a solution of a linear differential equation of the first order. Certain applications to hypergeometric series are given. The results allow to apply general theorems after V.G. Chirskii on the atrithmetic properties of the values of $F$-series. The result is that the values of the considered series at algebraic points, as well as at polyadic points, which are well approximable by rational integers, are infinitely algebraically independent. The paper also mentions some applications of polyadic and almost polyadic numbers to some practical problems.

Détail

Keywords: $F$ – series, infinite algebraic independence, polyadic numbers.

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