Infinite algebraic independence of some almost polyadic numbers

V. Y. Matveev

Čebyševskij sbornik, Tome 25 (2024) no. 3, pp. 365-372

Résumé

The paper considers $F$-series $f_{i,j}(z) = \sum_{n=0}^{\infty} {\left(\alpha_i\right)}_n{\left(\beta_j z \right)}^n$, where $\alpha_i$, $\beta_j$ are some rational numbers. These series satisfy a system of first-order linear differential equations with coefficients from $\mathbb{C}(z)$. Using previous results obtained using the approach proposed in one of the works of V.Kh. Salikhov, the algebraic independence of these series over $\mathbb{C}(z)$ is established. Application of the general theorem on the arithmetic properties of $F$-series from the works of V.G. Chirsky, allows us to assert the infinite algebraic independence of the values of these series. This means that for any polynomial $P\left(x_{1,1},\ldots,x_{m,n}\right)$ with integer coefficients other than the identical zero and any integer $\xi \ne 0 $, there is an infinite set of prime numbers $p$ such that in the field $\mathbb{Q}_p$ the inequality ${\left|P\left(f_{1,1}^{(p)}(\xi), \ldots,f_{m,n}^{(p)}(\xi)\right)\right|}_p \ne 0$. Here the symbols $f_{ij}^{(p)}\left(\xi\right)$ denote the sums of the series $\sum_{n=0}^{\infty}\left(\alpha_i\right)_n \left( \beta_j \xi\right)^n$ in the field $\mathbb{Q}_p$.

Détail

Keywords: Infinite algebraic independence, almost polyadic numbers.