\documentclass[12pt]{article}

\usepackage{amssymb,amscd,amsmath,amsthm,amsfonts}
\newcommand{\qq}{\mathfrak{q}}


\newcommand{\lcm}{\operatorname{lcm}}

\begin{document}

\begin{abstract}
Let $a_1,\ldots,a_k$ be positive integers generating the unit
ideal, and $j$ be a residue class modulo $L =
\lcm(a_1,\ldots,a_k)$. It is known that the function $r(N)$ that
counts solutions to the equation $x_1a_1 + \ldots + x_ka_k = N$ in
non-negative integers $x_i$ is a polynomial when restricted to
non-negative integers $N \equiv j \pmod L$.  Here we give, in the
case of $k=3$, exact formulas for these polynomials up to the
constant terms, and exact formulas including the constants for
$\qq = \gcd(a_1,a_2) \cdot \gcd(a_1,a_3) \cdot \gcd(a_2,a_3)$ of
the $L$ residue classes. The case $\qq = L$ plays a special
role, and it is studied in more detail.
%The paper is
%written for a general mathematical audience; in particular we have
%made an effort to replace (where possible)
%generatingfunctionological arguments with geometric and
%combinatorial considerations.
\end{abstract}

\end{document}
