Comparison of Polynomial-Oriented Computer Algebra Systems
Robert H Lewis
Michael Wester
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Abstract
[Now twelve years old, but still worth a look.] Exact symbolic computation with polynomials and matrices over polynomial rings has wide applicability to many fields. By "exact symbolic" we mean computation with polynomials whose coefficients are integers (of any size), rational numbers, or finite fields, as opposed to coefficients that are "floats" of a certain precision. Such computation is part of most computer algebra systems ("CA systems"). Over the last dozen years several large CA systems have become widely available, such as Axiom, Derive, Macsyma, Magma, Maple, Mathematica, and Reduce. They tend to have great breadth, be produced by profit-making companies, and be relatively expensive. However, most if not all of these systems have difficulty computing with the polynomials and matrices that arise in actual research. Real problems tend to produce large polynomials and large matrices that the general CA systems cannot handle. In the last few years several smaller CA systems focused on polynomials have been produced at universities by individual researchers or small teams. They run on Macs, PCs, and workstations. They are freeware or shareware. Several claim to be much more efficient than the large systems at exact polynomial computations. The list of these systems includes CoCoA, Fermat, MuPAD, Pari-GP, and Singular.
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