Arithmetic properties of direct product of $p$-adic fields elements,~II

A. S. Samsonov

Čebyševskij sbornik, Tome 22 (2021) no. 2, pp. 236-256.

Résumé

The article takes a look at transcendence and algebraic independence problems, introduces statements and proofs of theorems for some kinds of elements from direct product of $p$-adic fields and polynomial estimation theorem. Let $\mathbb{Q}_p$ be the $p$-adic completion of $\mathbb{Q}$, $\Omega_{p}$ be the completion of the algebraic closure of $\mathbb{Q}_p$, $g=p_1p_2\ldots p_n$ be a composition of separate prime numbers, $\mathbb{Q}_g$ be the $g$-adic completion of $\mathbb{Q}$, in other words $\mathbb{Q}_{p_1}\oplus\ldots\oplus\mathbb{Q}_{p_n}$. The ring $\Omega_g\cong\Omega_{p_1}\oplus\ldots\oplus\Omega_{p_n}$, a subring $\mathbb{Q}_g$, transcendence and algebraic independence over $\mathbb{Q}_g$ are under consideration. Here are appropriate theorems for numbers not only like $\alpha=\sum\limits_{j=0}^{\infty}a_{j}g^{r_{j}}$ where $a_{j}\in \mathbb Z_g,$ and non-negative rationals $r_{j}$ increase strictly unbounded. But, for numbers $f(\alpha)$, where $f(z)=\sum\limits_{j=0}^{\infty}c_jz^j\in\mathbb Z_g[[z]]$. Furthermore, let $\widehat{\mathbb Q}\cong\prod\limits_{p}\mathbb{Q}_p$ be the ring of polyadic numbers, then, the article takes a look at $\widehat{\Omega}=\prod\limits_{p}\Omega_p$, there are similar results for numbers like $f(\alpha)$, where $f(z)=\sum\limits_{j=0}^{\infty}c_jz^j\in\widehat{\mathbb Z}[[z]]$, $\alpha=\sum\limits_{k=1}^{\infty}a_{k}g^{r_{k}}$, $a_{k}\in \mathbb Z_g,$ $g=(p_1,\ldots,p_n,\ldots)$.

Détail

Keywords: $p$-adic numbers, $g$-adic numbers, polyadic numbers, transcendence, algebraic independence.

Bibliographie

[1] Adams W., “Transcendental numbers in the $p$-adic domain”, Amer. J. Math., 88 (1966), 279–307 | DOI | MR

[2] Amice Y., Les nombers $p$-adiques, Presses Universitaires de France, Paris, 1975 | MR

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

[4] Borevich Z. I., Shafarevich I. R., Teoriya chisel, 3-e izd. dop., Nauka, M., 1985

[5] Bundschuh P., Chirskii V. G., “On the algebraic independence of elements from $\mathbb C_p$ over $\mathbb Q_p$, I”, Arch. Math., 79 (2002), 345–352 | DOI | MR | Zbl

[6] Bundschuh P., Chirskii V. G., “On the algebraic independence of elements from $\mathbb C_p$ over $\mathbb Q_p$, II”, Acta Arithm., 113:4 (2004), 309–326 | DOI | MR | Zbl

[7] Bundschuh P., Chirskii V. G., “Estimating polynomials over $\mathbb Z_p$ at points from $\mathbb C_p$”, Moscow Journ. of Comb. and Number Th., 5:1-2 (2015), 14–20 | MR | Zbl

[8] Chirskii V. G., “Metod Zigelya-Shidlovskogo v r-adicheskoi oblasti”, Fundamentalnaya i prikladnaya matematika, 11:6 (2005), 221–230

[9] Chirskii V. G., “Values of Analytic functions at points of $\mathbb C_p$”, Russian Journ. of Math. Physics, 20:2 (2013), 149–154 | DOI | MR | Zbl

[10] Chirskii V. G., “Arifmeticheskie svoistva poliadicheskikh ryadov s periodicheskimi koeffitsientami”, Doklady akademii nauk, 459:6 (2014), 677–679 | DOI | Zbl

[11] Chirskii V. G., “Ob arifmeticheskikh svoistvakh obobschennykh gipergeometricheskikh ryadov s irratsionalnymi parametrami”, Izv. RAN. Ser. mat., 78:6 (2014), 193–210 | DOI | MR | Zbl

[12] Chirskii V. G., “Ob arifmeticheskikh svoistvakh ryada Eilera”, Vestnik Mosk. un-ta, Ser. 1, mat., mekh., 2015, no. 1, 59–61 | MR | Zbl

[13] Chirskii V. G., “Arifmeticheskie svoistva tselykh poliadicheskikh chisel”, Chebyshevskii sbornik, 16:1 (2015), 254–264 | MR | Zbl

[14] Chirskii V. G., “Arifmeticheskie svoistva poliadicheskikh ryadov s periodicheskimi koeffitsientami”, Izv. RAN. Ser. mat., 81:2 (2017), 215–232 | DOI | MR | Zbl

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

[16] Chirskii V. G., “Arifmeticheskie svoistva obobschennykh gipergeometricheskikh f-ryadov”, Doklady akademii nauk, 483:3 (2018), 257–259

[17] Chirskii V. G., “Arithmetic properties of generalized hypergeometric f-series”, Doklady Mathematics, 98:3 (2018), 589–591 | DOI | MR | Zbl

[18] Chirskii V. G., “Product formula, global relations and polyadic integers”, Russian Journ. of Math. Physics, 26:3 (2019), 286–305 | DOI | MR | Zbl

[19] Chirskii V. G., “Arithmetic properties of generalized hypergeometric series”, Russian Journ. of Math. Physics, 27:2 (2020), 175–184 | DOI | MR | Zbl

[20] Koblits N., $p$-adicheskie chisla, $p$-adicheskii analiz i dzeta-funktsii, per. s angl. V. V. Shokurova, ed. Yu. I. Manin, Mir, M., 1982 | MR

[21] Mahler K., “Uber transzendente $p$-adische Zahlen”, Compos. Math., 2 (1935), 259–275 | MR | Zbl

[22] Mahler K., $p$-adic numbers and their functions, second edition, Cambridge University Press, Cambridge, 1981 | MR | Zbl

[23] Shidlovskii A. B., Transtsendentnye chisla, Nauka, M., 1987