Infinite linear and algebraic independence of values of $F$-series at polyadic Liouvillea points

E. Yu. Yudenkova

Čebyševskij sbornik, Tome 22 (2021) no. 2, pp. 334-346

Résumé

This paper proves infinite linear and algebraic independence of the values of $F$-series at polyadic Liouville points using a modification of the generalised Siegel-Shidlovskii method. $F$-series have form $f_n = \sum_{n=0}^{\infty}a_n n! z^n$ whose coefficients $a_n$ satisfy some arithmetic properties. These series converge in the field $\mathbb{Q}_p$ of $p$-adic numbers and their algebraic extensions $\mathbb{K}_v$. Polyadic number is a series of the form $\sum_{n=0}^{\infty} a_nn!, a_n \in \mathbb{Z}$. Liouville number is a real number x with the property that, for every positive integer n, there exist infinitely many pairs of integers $(p, q)$ with $q > 1$ such that $0 \left| x - \frac{p}{q} \right| \frac{1}{q^n}. $ The polyadic Liouville number $\alpha$ has the property that for any numbers $P, D$ there exists an integer $|A|$ such that for all primes $p \leq P$ the inequality $|\alpha - A|_p A^{-D}. $ Infinite linear (algebraic) independence means that for any nonzero linear form (any nonzero polynomial) there are infinitely many primes $p$ and valuations $v$ extending $p$-adic valuation to an algebraic number field $\mathbb{K}$ with the following property: the result of substitution in the considered linear form (polynomial) of the values of $ F $ — of series instead of variables is a nonzero element of the field. Previously, only the existence of at least one prime number $p$ with the properties listed above was proved.

Détail

Keywords: Method by Siegel–Shidlovscii, $F$-series, polyadic Liouville numbers.