%%%%%%%%%%%%% start of file AlfPreamble.tex %%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %
% Alf van der Poorten
% ceNTRe for Number Theory Research
% Macquarie University
% NSW 2109 AUSTRALIA
% alf@mpce.mq.edu.au
%
%
% This file assumes the presence of AmSTeX 2.1 and font metrics 2.1 
% and the latest amsppt
% Without any AmSTeX give up. Otherwise remove the comment signs a few lines 
% below.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%\magnification=\magstep1
%\vsize 9.4truein
%\input amstex.tex	 %Make sure that it is v2.1, but see below.
%\documentstyle{amsppt}

% With earlier versions of AmSTeX and fonts you will get different breaks. 
% Don't blame me for the bad formatting. Various frills, like runningheads 
% mightn't work. Comment them out.


\parindent 0truepc
\parskip 4pt plus 2pt minus 2pt
\baselineskip 12pt \bigskip

\mathsurround=1pt


\def\C{{\Bbb C}} \def\N{{\Bbb N}} \def\L{{\Bbb L}} \def\K{{\Bbb K}} 
\def\Z{{\Bbb Z}} \def\Q{{\Bbb Q}} \def\A{\overline{\Bbb Q}} \def\R{{\Bbb R}}
\def\F{{\Bbb F}} %% means, with subscript, finite field to me 
\def\Fb{{\bold F}}

\def\({\bigl(} \def\){\bigr)}	%%Bloody useful for second parentheses
\def\\{\cr} %This may be a bad thing; check that it does no harm 
\def\lt{<}
\def\gt{>}


\def\ord{\mathop{\roman{ord}}\nolimits}
\def\trd{\mathop{\roman{transc\ deg}}\nolimits} 
\def\norm{\mathop{\roman{Norm}}\nolimits} 
\def\trace{\mathop{\roman{Trace}}\nolimits} 

\def\id#1{\langle #1\rangle} %%% ideal, in other words 
\def\larrow#1{\overleftarrow{#1}}
\def\rarr#1{\overrightarrow{#1}} 
\def\larr#1{\overleftarrow{#1}} 

\def\exactlydivides{\mathrel{\bigm\|}} 
\def\ediv{\exactlydivides} \def\divides{\mathrel{\bigm|}} 
\def\div{\divides}
\def\notdivides{\mathrel{\kern-3pt\not\!\kern3.5pt\bigm|}} 
\def\ndiv{\notdivides}

%%% A damned useful definition of \house for the size of algebraic numbers 
\newbox\boxA
\newbox\boxB
\newdimen\dimA
\newdimen\dimB
\newdimen\dimC

\def\house#1{%%%
\setbox\boxA\hbox{$\displaystyle #1$}%%% 
\dimA=1.04\wd\boxA\advance\dimA by 2pt
\dimB=1.04\ht\boxA\advance\dimB by 2pt
\dimC=0.05\wd\boxA
\hskip\dimC\hskip1pt\hbox to \dimA
{\vrule\vbox to \dimB{\hsize=\dimA
\hrule\vfill \line{\hfil \box\boxA\hfil}}\vrule}
\hskip 0.8pt\hskip1pt\hskip\dimC}



\def\poly {polynomial}
\def\ep {exponential polynomial}
\def\gps {generalised power sum}
\def\cf {continued fraction}
\def\pq {partial quotient}
\def\ap {approximation}
\def\ts{\textstyle }
\def\ds{\displaystyle }
\def\ex{expansion}
\def\cv {convergent}
\def\pf{paperfolding} %Watch it. This may conflict with LaTeX proofs 


%%%%For dealing with references at the draft stage 

\def\Ref#1{[#1]}
%\def\note#1{\relax}
\def\note#1{\footnote{#1}} 
\let\refnote=\note \def\label#1{\item{{[#1]}}}
\def\fl#1{\left\lfloor#1\right\rfloor}

%%%%Old black box signalling end of proof 
{\catcode`\@=11
\gdef\qed{\ifhmode\unskip\nobreak\fi\ifmmode\ifinner\else\hskip5\p@\fi\fi 
\hbox{\hskip5\p@\vrule width4\p@ height6\p@ depth1.5\p@\hskip\p@}} 

\newif\iflogo@
\gdef\nologo{\logo@false}
\logo@true
}

% Current time; this may be system dependent! 
\newcount\hours
\newcount\minutes
\def \SetTime{\hours=\time
\global\divide\hours by 60
\minutes=\hours
\multiply\minutes by 60
\advance\minutes by-\time
\global\multiply\minutes by-1 }
\SetTime
\def \now{\number\hours:\ifnum\minutes<10 0\fi\number\minutes} 
\def \Now{\today\ $[$\now$]$}



\font\dummy=dummy
%%%% My headlines
\font\headlinefont=cmti10 scaled 800
\font\vsmall=cmr6


\def\today{\ifcase\month\or
January\or February\or March\or April\or May\or June\or 
July\or August\or September \or October\or November\or December\fi 
\space\number\day,\space\number\year}


\def\ceNTRe{Centre for Number Theory Research} 

\def\myaddress{\noindent Alfred J. van der Poorten
\newline Centre for Number Theory Research
\newline Macquarie University\ \ NSW
2109\newline Australia\newline {\tt alf\@mpce.mq.edu.au}}



%%%%%%%%%%%%%%%%%%% Continued Fractions %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%These are not much good
\def\CF#1{{\def\\{\mathrel{,}}\def\;{\mathrel{;}}
\def\dotss{\dots\dots}[#1]}} 
\newtoks\CFbox
\newcount\CFlevel\CFlevel=0
\def\CFbox{\vtop{\hbox{\underbar{\CFnum}\CFext}\hbox{\CFnext}}} 
\def\CFraction#1#2#3#4{\def\CFnum{#1}\def\CFext{#2}\def\CFnext{#4}%
\ifnum\CFlevel=0\else\raise.6ex\fi%
\hbox{$#3\,+\;$\advance\CFlevel by1\CFbox}} \def\CFup{\raise.3ex}
%
\def\CFend#1{\CFup\hbox{$#1$}}	% for last term
\def\CFrac{\CFraction{\;1\;}{}}	% always 1 in the denominator.
\def\CFstop{~\lower.45ex\hbox{.}}	% correctly placed period.
\def\CFdots{\lower2.5ex\hbox{$\;\ddots$}} % correct diagonal dots. 
\def\CFautosub#1{#1_{\number\CFlevel}}
\def\aCFrac{\ifnum\CFlevel=0\setbox0=\hbox{$\;1\;$}\fi 
\CFraction{\CFup\copy0}{}{\ifnum\CFlevel=0 a_0\else\CFautosub{a}\fi}}
%\def\CFdisplay#1{{\def\;{\egroup\bgroup\CFrac\bgroup} 
%\def\\{ }\CFrac\bgroup#1}}}
%
%%Compare Cantor's suggestion
%finite continued fraction:
\def\cfrac#1#2{#1_0+{
\let\ds\displaystyle
\def\vl{\ds1\vrule width0pt depth.5ex height2ex\over} 
\vl{\ds#1_1 + {\vl{\ds#1_2 +{\vl\hskip.5em 
\ddots{\atop\hskip.5em{\vl\ds #1_{#2-1} +
{\vl\ds#1_{#2}}}}}}}}}}

%infinite continued fraction:
\def\cfraci#1#2{#1_0+{
\let\ds\displaystyle
\def\vl{\ds1\vrule width0pt depth.5ex height2ex\over} 
\vl{\ds#1_1 + {\vl{\ds#1_2 +{\vl\hskip.5em\ddots}}}}}} 
\comment
This is a finite continued fraction:
$$\cfrac am$$
This is an infinite continued fraction:
$$\cfraci am$$
\endcomment

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\date{Draft at \Now}\enddate
{\catcode`\@=11
\gdef\output@{\shipout\vbox{%
\iffirstpage@ \global\firstpage@false
\pagebody \logo@ \makefootline%
\else \ifrunheads@ \makeheadline\pagebody\makefootline 
\else \pagebody \makefootline \fi
\fi}%
\advancepageno \ifnum\outputpenalty>-\@MM\else\dosupereject\fi} 
}
\footline{\hfill \vsmall Revision of \Now} 

%
% Handling PostScript Graphics
%
%
% For use with Textures on a Macintosh ONLY!! %
\ifMacTextures

\gdef\EPSF#1by#2(#3){%
\vbox to #2{\hrule width #1 height 0pt depth 0pt% 
\vfill\special{illustration #3}}}%


\gdef\scaledEPSF#1by#2(#3 scaled #4){{%
\dimen0=#1 \dimen1=#2%
\divide\dimen0 by 1000 \multiply\dimen0 by #4% 
\divide\dimen1 by 1000 \multiply\dimen1 by #4% 
\EPSF \dimen0 by \dimen1 (#3 scaled #4)}}% 
\else
%
% For use with dvips by Tomas Rokicki, ONLY!! %
\input epsf

\gdef\EPSF#1by#2(#3){%
\vbox to #2{\hrule width #1 height 0pt depth 0pt\vfill \epsfbox{#3}}}%

\gdef\EPSF#1by#2(#3){\epsfbox{#3}}%

\gdef\scaledEPSF#1by#2(#3 scaled #4){{%
\dimen0=#1 \dimen1=#2%
\divide\dimen0 by 1000 \multiply\dimen0 by #4% 
\divide\dimen1 by 1000 \multiply\dimen1 by #4% 
\epsfxsize=\dimen0\epsfbox{#3}}}%
\fi


%This stuff is needed for the ceNTRe logo 
%The availability of the ps file logo.art 
%in the working directory is essential 

\def\centrelogo{\scaledEPSF 2.5in by 1.4in (logo.art scaled 650)} 

\def\smallcentrelogo{\scaledEPSF 2.5in by 1.4in (logo.art scaled 325)} 

\font \logofont=cmss8
\def\ceNTRelogo{\vtop{\baselineskip10truept
\hsize= 0.8125in \smallcentrelogo \logofont
\smallskip
\centerline{ceNTRe}
\centerline{Macquarie University}
\centerline{NSW 2109 Australia}} }

\def\Part=#1 {\def\part{#1}}


\endinput


%%%%%%%%%%%%% end of file AlfPreamble.tex %%%%%%%%%%%%%%%%