\documentclass[A4, 12pt, twoside]{article}
\usepackage{amsmath,amsfonts,amsthm,amssymb,amscd}
 \usepackage{amssymb}
 \usepackage[cp1251]{inputenc}
 \usepackage[T2A]{fontenc}
 \usepackage[all]{xy}
 \usepackage{latexsym}
 \usepackage{srcltx}
 \usepackage{amsmath}
 \usepackage{amsfonts}
 \usepackage{amscd}
 \usepackage{amssymb}
 \usepackage{amsthm}
 \usepackage{euscript}
 \usepackage{url}
 \usepackage{amssymb, amsthm}
 \usepackage{mathrsfs}
 \usepackage{hyperref}\usepackage{setspace}
 \usepackage[russian,english]{babel}
 \usepackage{mathrsfs}


 \newcommand{\refen}{\vspace{10mm} \noindent{\Large{\bf References}}}


 \sloppy
 \paperwidth=210mm
 \paperheight=297mm
 \textheight=250mm
 \textwidth=165mm
 \oddsidemargin=-0.4mm
 \evensidemargin=-5.4mm
 \topmargin=-5.4mm
 \footskip=12.5mm
 \headsep=0mm
 \headheight=0mm


 \begin{document}


\begin{center}
   \textbf{Indicial Rational Functions for Linear Ordinary Differential Equations
with Polynomial Coefficients \\[.3cm]
       S.S.~Ivanov}   (CC RAS, Moscow)
       
       A.A.~Petrova (MSU, Moscow)       
       \\
   \emph{E-mail address}: \texttt{ivanov@ccas.ru, petrova@yandex.ru}\\[.3cm]
\end{center}
\normalsize
We present an algorithm to construct an indicial rational
functions $V(z)$ for a differential equation
$$
a_n(z)y^{(n)}+\cdots +a_1(z)y'(z)+a_0(z)y(z)=f(z),
$$
where $f(z), a_i(z) \in {\bf K}[z]$, ${\bf K}$ is a field of charachteristic 0.
The algorithm is based on a balanced factorization $a_n(z)$ w.r.t. $f(z)$, $a_0(z)$, $a_1(z),\dots$, $a_{n-1}(z)$
followed by a sub-partition.

Our implementation is an improvement of the procedure {\bf ratsols} from the {\bf DEtools} package of the computer algebra system Maple.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%   Далее разместите аннотацию на русском языке   %%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{center}
   \textbf{Определяющие рациональные функции линейных обыкновенных уравнений с
полиномиальными коэффициентами\\[.3cm]
       С.С.~Иванов}   (ВЦ РАН, Москва)
       
       A.A.~Петрова (МГУ, Москва)       \\   \emph{E-mail address}: \texttt{ivanov@ccas.ru, petrova@yandex.ru}\\[.3cm]
\end{center}
\normalsize
Предлагается  алгоритм построения определяющей рациональной функции $V(z)$
для дифференцициального уравнения
$$
a_n(z)y^{(n)}+\cdots +a_1(z)y'(z)+a_0(z)y(z)=f(z),
$$
где $f(z), a_i(z) \in {\bf K}[z]$, ${\bf K}$ --- поле характеристики 0,
на  основе уравновешенной факторизации полинома $a_n(z)$ по отношению к  $f(z)$,  $a_0(z)$, $a_1(z),\dots$, $a_{n-1}(z)$
с последующим подразбиением.

Реализация  алгоритма  в системе Maple является улучшением существующей программы {\bf ratsols} из пакета  {\bf DEtools}.


\end{document}