Symbolic Computation versus Computer Algebra
Stephen M. Watt, University of Western Ontario
Friday, June 9, 2006, at U. of Waterloo.
Abstract:
We observe that "symbolic computation" and "computer algebra" are
really two different things and that neither one sufficiently
addresses the problems that arise in applications.
Symbolic computation may be seen as working with expression trees
representing mathematical formulae and applying various rules to
transform them. Computer algebra may be seen as developing
constructive algorithms to compute quantities in various arithmetic
domains, possibly involving indeterminates.
Symbolic computation allows a wider range of expression, but lacks
efficient algorithms. It is often unclear what is the algebraic structure
of a domain defined by rewrite rules. Computer algebra admits greater
algorithmic precision, but is limited in the problems that it can model.
We argue that considerable work is still required to make symbolic
computation more effective and computer algebra more expressive.
We use polynomials with symbolic exponents, e.g. $x^{n^2 + n}  y^{2m}$,
as an example that lies in the middle ground and we present algorithms
for their factorization and gcd.
