## 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 over a limited set of rings, or over 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 wellordering) and Mora's algorithm (if the ordering is a tangent cone ordering) as special cases. Furthermore, it provides polynomial factorization, 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. Schönemann; Humboldt-Universität 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 handle.

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 (with a much faster standard and Groebner bases computation based on Hilbert series and on an improved implementation of the core algorithms, libraries for primary decomposition, ring normalization, etc.)

For the highlights of the new SINGULAR version 4.4.1, see News and changes.

User manual for Singular version 4.4.1, 2025, generated by texi2html.