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, 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.
In this paper we compare these small systems to each other and to two of the large systems (Maple and Magma) on a set of problems involving exact symbolic computation with polynomials and matrices. The problems here involve:
Most of the actual code used in the benchmarks of the various systems is here also.