Macaulay2

Macaulay2 works with explicitly defined polynomial rings, ideals and modules for commutative algebra and algebraic geometry. The gb reference constructs a rational-coefficient ring, defines an ideal and computes a Gröbner-basis object. That returned object is not itself the generator matrix: use gens on it to obtain the sorted, auto-reduced basis columns. Algorithm selection and basis-element, pair or degree limits control the computation, so retain them with the ring and ideal definition. Input and output labels also distinguish a computed value from its type, which helps identify the mathematical object being passed to the next operation. The source is GPLv2-or-later; the repository states that distributed binaries are GPLv3 because of linked libraries. Native working and the linked university-hosted web interfaces have separate installation/access conditions.