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\chapter {Introduction to Macaulay}

\section{Background}

	\Macaulay\ is a system for computing in the fields of algebraic
geometry and commutative algebra; it is intended to provide a
computational research tool for working mathematicians.  The system
performs algebraic manipulation on polynomials, matrices, ideals,
polynomial rings, modules, maps between rings, and complexes of
modules. The possible manipulations include the computation of
standard (\Grobner) bases, modules of syzygies, finite free
resolutions, Hilbert polynomials and functions.  Using these basic
operations, a variety of derived operations are possible, such as
projections, ideal intersections, and the computation of coherent
sheaf cohomology groups.
 
	\Macaulay\ differs in a number of significant ways from other
computer algebra systems.  The computation of standard bases is its
fundamental operation, rather than simplification and factoring.
Unlike other systems which provide for the computation of standard
bases, submodules of free modules can uniformly be used wherever
ideals can be used.  \Macaulay\ is written in the language C and its
design has been optimized for execution on small systems; \Macaulay\ is
available, and reasonably powerful, on a Macintosh microcomputer.
Computations in \Macaulay\ are interruptable, with the full power of
the system available to study and use partial as well as complete
results of computations. \Macaulay\ is command-driven, and is quickly
learned by mathematicians having no experience with computers.

	\Macaulay\ has been continuously evolving since 1977. \Macaulay\
is currently implemented on the Macintosh and Sun, and has been used
by mathematicians in a number of countries, including Canada, France,
Germany, Italy, Norway, and the USA.

\section{Summary of New Features}

There are a number of new commands available with version 3.  
\begin {itemize}
  \item Monomial orders are more general in that the user can specify
an arbitrary monomial order.
  \item Parameters can be passed to command files, and a limited form of programming is allowed.
  \item Integer and polynomial expressions are now allowed on input.
  \item The standard bases commands are more flexible.
  \item Quotient rings have been added.
  \item Computations can be performed modulo a homogeneous ideal.
  \item A fast codimension and degree algorithm has been added.
  \item A large set of scripts (command files) are provided.  These scripts 
were designed and implemented by David Eisenbud and Mike Stillman.
\end {itemize} 

\section{Conventions}

Throughout this document the operating system prompt is represented by
a ``\#''.  The \Macaulay system prompt is represented by ``\%''.  All
\Macaulay\ dialogs are displayed in a typewriter font.

Some commands require parameters while others have optional
parameters.  Required parameters are enclosed with angle brackets i.e.

\cmd{command} {\parm {required argument1}  \ldots \parm {required 
  argument n} }

Optional parameters are enclosed in square brackets i.e.:

\cmd{command} {\optparm {optional argument1}  \ldots \optparm {optional 
  argument n} }

	Commands which have default parameters will use the default
value if the user enters a carriage return.  We will denote this by
\return.  (On the Macintosh, one uses the {\tt <enter>} key, see 
\appref{ap-mpw}).


\section{Acknowledgements}

There are two people who have had a large influence on us as well as on
\Macaulay.  Graham Evans first introduced one of us (Mike Stillman) to
the wonders of syzygies and finite free resolutions, which led to the 
ancestor of \Macaulay\ written in the late 1970's.  
Without him, this system would not exist.  His continual support has been
invaluable.

David Eisenbud has also played a large role through the years.  His
support and suggestions have helped shape the entire project, and have
been invaluable.  He wrote a large majority of the scripts in
\appref{ap-scripts}.  This large amount of work has made \Macaulay\ a
much more powerful system.

We would also like to thank the Mathematical Sciences Insitute at Cornell,
for providing us with the Macintosh MPW license, as well as computer
equipment and hospitality.

