On Gol'dberg's characteristics for multiple power series (Q1605582)

Let \(f(z)=\sum_{k\in\mathbb Z_+^n}a_kz^k=\sum_{k_1,\dots,k_n=0}^{\infty} a_{k_1,\dots,k_n}z_1^{k_1}\cdots z_n^{k_n}\) be an entire in \(\mathbb C^n\) function. \textit{A. A. Gol'dberg} and \textit{I. F. Bitlyan} [Vestn. Leningr. Univ. 14, No. 13 (Ser. Mat. Mekh. Astron. No. 3), 27-41 (1959; Zbl 0094.05503)] considered for such a function besides the maximum modulus \(M(r)\) and maximum term \(m(r)=\max\{|a_k|r^k: k\in\mathbb Z_+^n\}\) the system of central indices \(\nu(r)=(\nu_1(r),\dots,\nu_n(r))\), \(\nu_j(r)=\max\{k_j:k\in{\mathbb Z}_+^n;|a_k|r^k=m(r)\}\), \(j=1,\dots,n\) and generalized the well-known Wiman-Valiron theorems. In \textit{A. A. Gol'dberg} [Dokl. Akad. Nauk Arm. SSR 29, 145-151 (1959; Zbl 0090.05404); Dokl. Uzhgorod. Univ. 4, 101-103 (1961)] new growth characteristics for entire functions of several variables were defined, namely the \((G,\rho_1,\dots,\rho_n)\)-orders, the \((G,\rho_1,\dots,\rho_n)\)-types, and also the systems of \(G\)-orders and \(G\)-types. In these papers formulas for the calculation of the mentioned characteristics are derived by means of the Taylor coefficients of the functions. In the paper under review the author describes the main properties of the mentioned characteristics.