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 Chebyshevskii Sb., 2017, Volume 18, Issue 4, Pages 256–260 (Mi cheb609)

Estimates of polynomials in a liouvillean polyadic integer

E. S. Krupitsyn

Moscow State Pedagogical University

Abstract: Let
$$\alpha=\sum\limits_{n=0}^\infty a_kn_k!, \quad a_k\in\mathbb{Z}, \quad 0\leqslant a_k\leqslant n_k,$$
with a rapidly growing sequence $n_k$ of positive integers. This series converges in all $p$-adic fields $\mathbb{Q}_p$ so it is a polyadic number.
The ring of polyadic integers is a direct product of the rings $\mathbb{Z}_p$ of $p$-adic integers over all prime numbers $p$.
So $\alpha$ can be considered as the vector $(\alpha^{(1)}, \ldots, \alpha^{(n)}, \ldots)$ with coordinates equal to the sums $\alpha^{(n)}$ of the series $\alpha$ in the field $\mathbb{Q}_{p_n}$ for the $n$-th prime $p_n$.
For any nonzero polynomial $P(x)$ with integer coefficients one has
$$P(\alpha)=(P(\alpha^{(1)}), \ldots, P(\alpha^{(n)}), \ldots ).$$

The polyadic integer $\alpha$ is called transcendental, if for any nonzero polynomial $P(x)$ with rational integer coefficients there exist a prime $p^{(n)}$ with $P(\alpha^{(n)})\neq 0$ in $p_n$.
The polyadic integer is infinitely transcendental if there exist infinitely many primes $p_n$ such that $P(\alpha^{(n)})\neq 0$ in $\mathbb{Q}_{p_n}$ and it is called globally transcendental, if $P(\alpha^{(n)})\neq 0$ for any $n$.
The paper presents estimates from below of $|P(\alpha^{(n)})|_{p_n}$ in any $\mathbb{Q}_{p_n}$. As a corollary we get the global transcendence of $\alpha$.

Keywords: polyadic integer, estimates of polynomials.

DOI: https://doi.org/10.22405/2226-8383-2017-18-4-255-259

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UDC: 517
Accepted:15.12.2017

Citation: E. S. Krupitsyn, “Estimates of polynomials in a liouvillean polyadic integer”, Chebyshevskii Sb., 18:4 (2017), 256–260

Citation in format AMSBIB
\Bibitem{Kru17} \by E.~S.~Krupitsyn \paper Estimates of polynomials in a liouvillean polyadic integer \jour Chebyshevskii Sb. \yr 2017 \vol 18 \issue 4 \pages 256--260 \mathnet{http://mi.mathnet.ru/cheb609} \crossref{https://doi.org/10.22405/2226-8383-2017-18-4-255-259} \elib{http://elibrary.ru/item.asp?id=30042558}