Symbolic Computation versus Computer Algebra

Stephen M. Watt
Ontario Research Centre for Computer Algebra
University of Western Ontario
London, Canada 


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.