 There should be one TeX file, that is, the references should already
be in the TeX file (no bibfile!), and please no separate
files for each section! Of course, if you have separate figure files, this
is totally okay.

The article must be in 12pt.

The paper size should not be exceedingly large.
(textwidth 16cm=6.3in and textheight 23cm=9in are maxima.)

Replace \date{...} by \date{}.
(No date should show up.)

Look through the file whether you discover any
>From
(as a result of email transmission.)

Run the file through a spelling checker.
Experience shows that it always finds something.

If you write in English, be aware of conventions and meanings
of words in English: for example,

There are no mathematical objects called "applications" in
English. Instead, there are "maps".

It is not advisable to develop a function f(x) in
a Taylor series, but it is preferable to expand f(x)
in a Taylor series, respectively to compute its Taylor series
expansion.

The terms "positive" and "negative" do not mean the same in
English as they do in French.

The French verb "remarquer" can have two different translations
in English: "to remark" (in case you want to tell something to the
reader) or "to observe" (in case that you "note" something).

One cannot "precise" anything. Instead, one can "make something
precise".

Similarly, one cannot "explicit" anything.
Instead, one can "make something explicit".

In a similar vein, one cannot "explicate" anything. Instead, one can
"explain" something (respectively "make something
explicit," depending on what you really want to say).

"analogue" is exclusively a substantive (and it is the
nonAmerican spelling). The adjective is "analogous".

"The Macdonald's theorem" does not exist. Instead there is
"the theorem of Macdonald" or "Macdonald's theorem" (no "the" or "a"
before the latter!).

Do not use abbreviations such as "iff", "rhs", "resp.", "w.r.t.", etc.
Use "if and only if", "righthand side", "respectively", "with respect to",
etc.,
instead.

If you use LaTeX (as, of course, most do nowadays), then, please,
do not use LaTeX's "article" format
\documentclass[12pt]{article}
It looks terrible with all these huge white spaces and letters.
Instead we recommend the AmSLaTeX article format
\documentclass[12pt]{amsart}
(unless you have something even more beautiful).
 In order to avoid inappropiate line breaks, one should use "~" wherever
it is necessary. For example, one should write
Section~5
instead of "Section 5", and one should write
Theorem~\ref{...}
instead of "Theorem \ref{...}", etc.

In general, do not use
\begin{eqnarray}
\end{eqnarray}
It looks terrible. Instead use
\begin{align}
\end{align}
or
\begin{multline}
\end{multline}
or
\begin{gather}
\end{gather}

Do not use "align" in the form
\begin{align}
LHS1 =& RHS1 \\
LHS2 =& RHS2 \\
...
\end{align}
it generates a wrong spacing. Instead use it in the form
\begin{align}
LHS1 &= RHS1 \\
LHS2 &= RHS2 \\
...
\end{align}

The reason of warnings of the kind
Package amsmath Warning: Foreign command \over;
(amsmath) \frac or \genfrac should be used instead
(amsmath) on input line 150.
is that, in AmSLaTeX, one should use
\frac {numerator} {denominator}
instead of
{numerator \over denominator} ,
respectively
\binom {upper} {lower}
instead of
{upper \choose lower} ,
respectively
\begin{bmatrix} upper \\ lower \end{bmatrix}
instead of
{upper \brack lower} .

Follow a convention, and follow only one convention, when you
write the references. It is not important which convention you choose,
but, please, do not write things like
1. I.G. Macdonald, Affine root systems and Dedekind's etafunction
Invent. Math. 15 (1972), 91143.
2. Macdonald, I. G. Some conjectures for root systems,
SIAM J. Math. Anal. 13:9881007, 1982.
(This makes at least 5 inconsistencies.)