### This paper was presented as a poster to the 1999 ISSAC Conference, Vancouver, British Columbia, 28-31 July 1999.

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:

the ground rings Z, Q, Z/p, 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

determinant, characteristic polynomial

gcd of multivariate polynomials

resultants

Most of the actual code used in the benchmarks of the various systems is here also.

rlewis@fordham.edu

wester@math.unm.edu