Macaulay computer algebra system
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Macaulay is a computer algebra system for doing polynomial computations, particularly Gröbner basis calculations. Macaulay is designed for solving problems in algebraic geometry and has a quite simple syntax, often described as "algebraic machine language". Two incarnations exist, Macaulay and Macaulay2.
Macaulay is freely downloadable with source, and is available for Macintosh, Linux, and Windows.
Macaulay2 is a rewrite of Macaulay and is published under the GPL.