2.1 Background
==============

SINGULAR is a Computer Algebra system for polynomial
computations with emphasis on the special needs of commutative
algebra, algebraic geometry, and singularity theory.

SINGULAR's main computational objects are ideals and
modules over a large variety of baserings. The baserings are polynomial
rings or localizations thereof over a field (e.g., finite fields, the
rationals, floats, algebraic extensions, transcendental extensions) or
quotient rings with respect to an ideal.

SINGULAR features one of the fastest and most general
implementations of various algorithms for computing Groebner
resp. standard bases. The implementation includes Buchberger's algorithm
(if the ordering is a well ordering) and Mora's algorithm (if the
ordering is a tangent cone ordering) as special cases.  Furthermore, it
provides polynomial factorizations, resultant, characteristic set and
gcd computations, syzygy and free-resolution computations, and many more
related functionalities.

Based on an easy-to-use interactive shell and a C-like programming
language, SINGULAR's internal functionality is augmented and
user-extendible by libraries written in the SINGULAR programming
language. A general and efficient implementation of communication links
allows SINGULAR to make its functionality available to other
programs.

SINGULAR's development started in 1984 with an implementation of
Mora's Tangent Cone algorithm in Modula-2 on an Atari computer (K.P.
Neuendorf, G. Pfister,
H.\ Sch\"onemann; Humboldt-Universit\"at
 zu Berlin).  The need for a new system arose from the investigation of
mathematical problems coming from singularity theory which none of the
existing systems was able to compute.

In the early 1990s SINGULAR's "home-town" moved to
Kaiserslautern, a general standard basis algorithm was implemented in C
and SINGULAR was ported to Unix, MS-DOS, Windows NT, and MacOS.

Continuous extensions (like polynomial factorization, gcd computations,
links) and refinements led in 1997 to the release of SINGULAR
version 1.0 and in 1998 to the release of version 1.2
(much faster standard and Groebner bases computations based on Hilbert series
and on improved implementations of the algorithms,
libraries for primary decomposition, ring normalization, etc.)

For the highlights of the new SINGULAR version 2-0-4 see News and changes.


<font size="-1">
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; User manual for <a href="http://www.singular.uni-kl.de/"><i>Singular</i></a> version 2-0-4, October 2002,
generated by <a href="http://www.gnu.org/software/texinfo/"><i>texi2html</i></a>.
</font>

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