CoCoA

CoCoA
is a free computer algebra system developed by the University of Genova, Italy, used to compute with numbers and polynomials. The CoCoA Library
is available under GNU General Public License. CoCoA has been ported to many operating systems including Macintosh on PPC and x86, Linux on x86, Unix x86-64 & PPC, Solaris on SPARC and Windows on x86.
CoCoA is mainly used by researchers,
but can be useful even for "simple" computations.
CoCoA's features include:

Very big integers and rational numbers using the GNU Multi-Precision Library
Multivariate Polynomials
Gröbner basis
User interfaces: text; Emacs-based; Qt-based

It is able to perform simple and sophisticated operations on multivariate polynomials and on various data related to them. For example, it can readily compute Gröbner basis, syzygies and minimal free resolutions, intersection, division, the radical of an ideal, the ideal of zero-dimensional schemes, Poincaré series and Hilbert functions, factorization of polynomials, and toric ideals. The capabilities of CoCoA and the flexibility of its use are further enhanced by the dedicated high-level programming language.
Its mathematical core, CoCoALib, has been designed as an open source C++ library, focussing on ease of use and flexibility.
It is available on GitHub: at https://github.com/cocoa-official/CoCoALib
CoCoALib is based on GNU Multi-Precision Library.
CoCoALib is used by
ApCoCoA
and
NmzIntegrate