Algebraic independence of certain almost polyadic series

V. Yu. Matveev

Čebyševskij sbornik, Tome 17 (2016) no. 3, pp. 166-177.

Résumé

The paper describes the arithmetic nature of the values at integer points of series from the so-called class of $F$-series which constitute a solution of a system of linear differential equations with coefficients — rational functions in $z$. We consider a subclass of the series consisting of the series of the form \begin{equation} \nonumber \sum_{n=0}^\infty a_n\cdot n!\; z^n \end{equation} where $a_n\in\mathbb Q$, $|a_n|\leq e^{c_1 n}$, $n=0,1,\ldots$ with some constant $c_1$. Besides there exists a sequence of positive integers $d_n$ such that $d_n\; a_k\in\mathbb Z$, $k=0,\ldots,n$ and $d_n=d_{0,n} d_n$, $d_{0,n}\in\mathbb N$, $n=0,1,\ldots,d\in\mathbb N$ and for any $n$ the number $d_{0,n}$ is divisible only by primes $p$ such that $p\leqslant c_2 n$. Moreover \begin{equation} \nonumber ord_p n \leq c_3\left(\log_p n+\frac{n}{p^2}\right). \end{equation} We say then that the considered series belongs to the class $F(\mathbb{Q},c_1,c_2,c_3,d)$. Such series converge at a point $z\in\mathbb Z$, $z\ne 0$ in the field $\mathbb Q_p$ for almost all primes $p$. The direct product of the rings $\mathbb Z_p$ of $p$-adic integers over all primes $p$ is called the ring of polyadic integers. It's elements have the form \begin{equation} \nonumber \mathfrak{a} = \sum_{n=0}^\infty a_n\cdot n!,\quad a_n\in\mathbb Z \end{equation} and they can be considered as vectors with coordinates $\mathfrak{a}^{(p)}$ which are equal to the sum of the series $\mathfrak{a}$ in the field $\mathbb Q_p$ (This direct product is infinite). For any polynomial $P(x)$ with integer coefficients we define $P(\mathfrak{a})$ as the vector with coordinates $P(\mathfrak{a}^{(p)})$ in $\mathbb Q_p$. According to the classification, described in V. G. Chirskii's works we call polyadic numbers $\mathfrak{a}_1,\ldots,\mathfrak{a}_m$ infinitely algebraically independent, if for any nonzero polynomial $P(x_1,\ldots,x_m)$ with integer coefficients there exist infinitely many primes $p$ such that \begin{equation} \nonumber P\left(\mathfrak{a}_1^{(p)},\ldots,\mathfrak{a}_m^{(p)}\right)\ne 0 \end{equation} in $\mathbb Q_p$. The present paper states that if the considered $F$-series $f_1,\ldots,f_m$ satisfy a system of differential equations of the form \begin{equation} \nonumber P_{1,i}y_i^\prime + P_{0,i}y_i = Q_i, i=1,\ldots,m \end{equation} where the coefficients $P_{0,i}, P_{1,i}, Q_i$ are rational functions in $z$ and if $\xi\in\mathbb Z$, $\xi\ne 0$, $\xi$ is not a pole of any of these functions and if \begin{equation} \nonumber \exp\left(\int\left(\frac{P_{0,i}(z)}{P_{1,i}(z)}-\frac{P_{0,j}(z)}{P_{1,j}(z)}\right)dz\right)\not\in\mathbb C(z) \end{equation} then $f_1(\xi),\ldots,f_m(\xi)$ are infinitely algebraically independent almost polyadic numbers. For the proof we use a modification of the Siegel–Shidlovsky's method and V. G. Chirskii's. Salikhov's approach to prove the algebraic independence of functions, constituting a solution of the above system of differential equations. Bibliography: 30 titles.

Détail

Keywords: algebraic independence, almost polyadic numbers.

Bibliographie

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

[2] 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

[3] Chirskii V. G., “Arithmetic properties of polyadic integers”, Tchebyshevskiy sbornik, 16:1 (2015), 254–264 | MR

[4] Shidlovskii A. B., Transcendental numbers, W. de Gruyter, 1989 | MR | MR | Zbl

[5] Salikhov V. Kh., “On algebraic independence of the values of E-functions which satisfy first order linear differential equations”, Mat. Zametki, 13:1 (1973), 29–40 | MR

[6] Chirskii V. G., “Global relations”, Math. Notes, 48 (1990), 795–798 | DOI | MR | Zbl

[7] Nesterenko Yu. V., “Hermite–Pade approximations of generalized hypergeometric functions”, Russ. Acad. Sci. Sb. Math., 85 (1995), 189–219 | MR | MR

[8] Chirskii V. G., “On the arithmetic properties Euler series”, Vestnik Mosc. Univ., 2015, no. 1, 59–61 (Russian)

[9] Postnikov A. G., Introduction to analytic number theory, Nauka, M., 1971, 416 pp. (Russian) | MR

[10] Pontryagin L. S., Continious groups, Nauka, M., 1984, 529 pp. (Russian) | MR

[11] Novoselov E. V., “Topological theory of divisibility”, Uchen. zapiski Elabugh. Ped. Inst., 3 (1960), 3–23

[12] Chirskii V. G., Shakirov R. F., “On representations of integers in DBNS”, Tchebyshevskiy sbornik, 14:1 (2013), 86–93 (Russian) | MR

[13] Matveev V. Yu., Chirskii V. G., “On a series of products of terms of an arithmetic progression”, Prepodavatel 21 veka, 2:4 (2013), 249–254 (Russian)

[14] Matveev V. Yu., “On the values of a certain series at polyadic points, well approximable by positive integers”, Prepodavatel 21 veka, 2:4 (2013), 255–259 (Russian) | Zbl

[15] Chirskii V. G., Matveev V. Yu., “On certain properties of polyadic expansions”, Tchebyshevskiy sbornik, 14:2 (2013), 164–172 (Russian)

[16] Chirskii V. G., “Estimates of linear forms and polynomials in polyadic integers”, Tchebyshevskiy sbornik, 12:4 (2011), 129–134 (Russian)

[17] Chirskii V. G., “Polyadic estimates for $F$-series”, Tchebyshevskiy sbornik, 13:2 (2012), 131–136 (Russian)

[18] Chirskii V. G., Matveev V. Yu., “On a representation of positive integers”, Tchebyshevskiy sbornik, 14:1 (2013), 92–101 (Russian)

[19] Chirskii V. G., Matveev V. Yu., “On a representation of positive integers”, Vestnik Mosc. Univ., 2013, no. 6, 57–59 (Russian)

[20] Chirskii V. G., “On the arithmetic properties of generalized hypergeometric series with irrational parameters”, Izvestiya: Mathematics, 78:6 (2014), 1244–1260 | DOI | DOI | MR | Zbl

[21] Chirskii V. G., “On nontrivial global relations”, Vestnik Mosc. Univ., 1989, no. 5, 33–36 (Russian)

[22] Chirskii V. G., “On algebraic relations in local fields”, Vestnik Mosc. Univ., 1990, no. 3, 92–95 (Russian)

[23] Chirskii V. G., “Global relations and hypergeometric series”, Uspekhi Mat. Nauk, 46:6(282) (1991), 221–222

[24] Chirskii V. G., “On algebraic relations in non-archimedian fields”, Funct. Anal. Appl., 26 (1992), 108–115 | DOI | MR | Zbl

[25] Chirskii V. G., “On series which are algebraically independent in all local fields”, Vestnik Mosc. Univ., 1994, no. 3, 93–95 (Russian)

[26] Chirskii V. G., “Estimates of polynomials and linear forms in direct products of fields”, Vestnik Mosc. Univ., 1994, no. 4, 35–39 (Russian)

[27] Chirskii V. G., Bundschuh P., “Algebraic independence of elements from $\mathbb{C}_p$ over $\mathbb{Q}_p$, II”, Acta Arithmetica, 113:4 (2004), 309–326 | DOI | MR | Zbl

[28] Chirskii V. G., “Siegel's method in $p$-adic domain”, Fundam i prikl. matem., 11:6 (2005), 221–230

[29] Chirskii V. G., “A generalization of the notion of global relation”, Zapiski nauch. Semin POMI, 322 (2005), 220–238 | MR | Zbl

[30] Chirskii V. G., Bundschuh P., “Algebraic independence of elements from $\mathbb{C}_p$ over $\mathbb{Q}_p$”, Arch. der Math., 79 (2002), 345–352 | DOI | MR | Zbl