Comparison of polynomial-oriented computer algebra systems

Exact symbolic computation with polynomials and matrices over polynomial rings has wide applicability to many fields [Hereman96, Lewis99]. By "exact symbolic", we mean computation with polynomials whose coefficients are integers (of any size), rational numbers, or from finite fields, as opposed to coefficients that are "floats" of a certain precision. Such computation is part of most computer algebra (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, at least for a full blown non-student version. 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 [Lewis99].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 [CoCoA, Fermat, MuPAD, Pari-Gp, Singular].In this paper, we compare these small systems to each other and to two of the large systems (Magma and Maple) on a set of problems involving exact symbolic computation with polynomials and matrices. The problems here involve:• the ground rings Z, Q, Z/p and other finite fields• basic arithmetic of polynomials over the ground ring• basic arithmetic of rational functions over the ground ring• polynomial evaluation (substitution)• matrix normal forms• determinants and characteristic polynomials• GCDs of multivariate polynomials• resultants