Estimates of polynomials in a liouvillean polyadic integer

E. S. Krupitsyn

Čebyševskij sbornik, Tome 18 (2017) no. 4, pp. 256-260.

Résumé

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 $\left(\alpha^{(1)}, \ldots, \alpha^{(n)}, \ldots\right)$ 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 $$ P(\alpha)=\left(P\left(\alpha^{(1)}\right), \ldots, P\left(\alpha^{(n)}\right), \ldots \right). $$ 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 $P\left(\alpha^{(n)}\right)\neq 0$ in $p_n$. The polyadic integer is infinitely transcendental if there exist infinitely many primes $p_n$ such that $P\left(\alpha^{(n)}\right)\neq 0$ in $\mathbb{Q}_{p_n}$ and it is called globally transcendental, if $P\left(\alpha^{(n)}\right)\neq 0$ for any $n$. The paper presents estimates from below of $\left|P\left(\alpha^{(n)}\right)\right|_{p_n}$ in any $\mathbb{Q}_{p_n}$. As a corollary we get the global transcendence of $\alpha$.

Détail

Keywords: polyadic integer, estimates of polynomials.

Bibliographie

[1] Postnikov A. G., Introduction to the analytic number theory, M., 1971

[2] Chirskii V. G., “Arithmetic properties of polyadic series with periodic coefficients”, Doklady Mathematics, 90:3 (2014), 766–768 | DOI | DOI | MR | Zbl

[3] Chirskii V. G., “Arithmetic properties of polyadic series with periodic coefficients”, Izvestiya Mathematics, 81:2 (2017), 444–461 | DOI | DOI | MR | Zbl

[4] Chirskii V. G., “An approach to the transformation of periodic sequences”, Chebushevskii sb., 17:3 (2016), 180–185 | DOI

[5] Chirskii V. G., “On the arithmetic properties of generalized hypergeometric series with irrational parameters”, Izvestiya Mathematics, 2014, no. 6, 1244–1260 | DOI | DOI | MR | Zbl

[6] Chirskii V. G., “On global relations”, Math. notes, 48:2 (1990), 123–127 | DOI | MR | Zbl

[7] Chirskii V. G., “Arithmetic properties of Euler series”, Moscow University Mathematics Bulletein, 70:1 (2015), 41–43 | DOI | MR | Zbl

[8] Chirskii V. G., Nesterenko A. Yu., “An approach to the transformation of periodic sequences”, Discrete Mathematics and Applications, 27:1 (2017), 1–6 | DOI | DOI | MR | Zbl

[9] Chirskii V. G., “On series which are algebraically independent in all local fields”, Vestnik Mosc. Univ., Ser. 1, math.-mech., 1994, no. 3, 93–95

[10] Chirskii V. G., “Estimates for linear forms and polynomials in polyadic numbers”, Tchebyshev. Sbornik, 12:4 (2011), 129–134

[11] Chirskii V. G., “Polyadic estimates for $F$-series”, Tchebyshev. Sbornik, 13:2 (2012), 131–136

[12] Bertrand D., Chirskii V. G., Yebbou Y., “Effective estimates for global relations on Euler-type series”, Ann. Fac. Sci. Toulouse, XIII:2 (2004), 241–260 | DOI | MR | Zbl

[13] Krupitsyn E. S., Chirskii V. G., “Estimates of polynomials in some polyadic numbers”, Prepodavatel' XXI century, 2012, no. 4, 217–224

[14] Cijsouw P. L., Transcendence measures, Acad. Proefschrift, Amsterdam, 1972 | MR

[15] Chirskii V. G., “Topical problems of the theory of transcendental numbers: Developments of approaches to their solutions in the works of Yu.V. Nesterenko”, Russian Journal of Mathematical Physics, 24:2 (2017), 153–171 | DOI | MR | Zbl