Singular 3-1-6

Singular  is a computer algebra system for polynomial computations, with special emphasis on commutative and non-commutative algebra, algebraic geometry, and singularity theory. It is free and open-source under the GNU General Public Licence.

Singular provides

    * highly efficient core algorithms,
    * a multitude of advanced algorithms in the above fields,
    * an intuitive, C-like programming language,
    * easy ways to make it user-extendible through libraries, and
    * a comprehensive online manual and help function.

Its main computational objects are ideals, modules and matrices over a large number of baserings. These include

    * polynomial rings over various ground fields and some rings (including the integers),
    * localizations of the above,
    * a general class of non-commutative algebras (including the exterior algebra and the Weyl algebra),
    * quotient rings of the above,
    * tensor products of the above.

Singular`s core algorithms handle

    * Gröbner resp. standard bases and free resolutions,
    * polynomial factorization,
    * resultants, characteristic sets, and numerical root finding.

Its advanced algorithms, contained in currently more than 90 libraries, address topics such as absolute factorization, algebraic D-modules, classification of singularities, deformation theory, Gauss-Manin systems, Hamburger-Noether (Puiseux) development, invariant theory, (non-) commutative homological algebra, normalization, primary decomposition, resolution of singularities, and sheaf cohomology.