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Algebraic independence of certain almost polyadic series

V. Yu. Matveev


Abstract: The paper describes the arithmetic nature of the values at integer points of series from the so-called class of $F$-series which constitute a solution of a system of linear differential equations with coefficients — rational functions in $z$.
We consider a subclass of the series consisting of the series of the form
\begin{equation} \nonumber \sum_{n=0}^\infty a_n\cdot n!\; z^n \end{equation}
where $a_n\in\mathbb Q$, $|a_n|\leq e^{c_1 n}$, $n=0,1,\ldots$ with some constant $c_1$. Besides there exists a sequence of positive integers $d_n$ such that $d_n\; a_k\in\mathbb Z$, $k=0,\ldots,n$ and $d_n=d_{0,n} d_n$, $d_{0,n}\in\mathbb N$, $n=0,1,\ldots,d\in\mathbb N$ and for any $n$ the number $d_{0,n}$ is divisible only by primes $p$ such that $p\leqslant c_2 n$. Moreover
\begin{equation} \nonumber ord_p n \leq c_3\left(\log_p n+\frac{n}{p^2}\right). \end{equation}
We say then that the considered series belongs to the class $F(\mathbb{Q},c_1,c_2,c_3,d)$. Such series converge at a point $z\in\mathbb Z$, $z\ne 0$ in the field $\mathbb Q_p$ for almost all primes $p$.
The direct product of the rings $\mathbb Z_p$ of $p$-adic integers over all primes $p$ is called the ring of polyadic integers. It's elements have the form
\begin{equation} \nonumber \mathfrak{a} = \sum_{n=0}^\infty a_n\cdot n!,\quad a_n\in\mathbb Z \end{equation}
and they can be considered as vectors with coordinates $\mathfrak{a}^{(p)}$ which are equal to the sum of the series $\mathfrak{a}$ in the field $\mathbb Q_p$ (This direct product is infinite).
For any polynomial $P(x)$ with integer coefficients we define $P(\mathfrak{a})$ as the vector with coordinates $P(\mathfrak{a}^{(p)})$ in $\mathbb Q_p$. According to the classification, described in V. G. Chirskii's works we call polyadic numbers $\mathfrak{a}_1,\ldots,\mathfrak{a}_m$ infinitely algebraically independent, if for any nonzero polynomial $P(x_1,\ldots,x_m)$ with integer coefficients there exist infinitely many primes $p$ such that
\begin{equation} \nonumber P\left(\mathfrak{a}_1^{(p)},\ldots,\mathfrak{a}_m^{(p)}\right)\ne 0 \end{equation}
in $\mathbb Q_p$.
The present paper states that if the considered $F$-series $f_1,\ldots,f_m$ satisfy a system of differential equations of the form
\begin{equation} \nonumber P_{1,i}y_i^\prime + P_{0,i}y_i = Q_i, i=1,\ldots,m \end{equation}
where the coefficients $P_{0,i}, P_{1,i}, Q_i$ are rational functions in $z$ and if $\xi\in\mathbb Z$, $\xi\ne 0$, $\xi$ is not a pole of any of these functions and if
\begin{equation} \nonumber \exp\left(\int\left(\frac{P_{0,i}(z)}{P_{1,i}(z)}-\frac{P_{0,j}(z)}{P_{1,j}(z)}\right)dz\right)\not\in\mathbb C(z) \end{equation}
then $f_1(\xi),\ldots,f_m(\xi)$ are infinitely algebraically independent almost polyadic numbers.
For the proof we use a modification of the Siegel–Shidlovsky's method and V. G. Chirskii's. Salikhov's approach to prove the algebraic independence of functions, constituting a solution of the above system of differential equations.
Bibliography: 30 titles.

Keywords: algebraic independence, almost polyadic numbers.

UDC: 511.36

Received: 30.06.2016
Accepted: 13.09.2016

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