Making Computer Algebra More Symbolic

This paper is a step to bring closer together two views of computing with mathematical objects: the view of “symbolic computation” and the view of “computer algebra.” 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 algebraic quantities in various arithmetic domains, possibly involving indeterminates. Symbolic computation allows a wider range of expression, while computer algebra admits greater algorithmic precision. We examine the problem of providing polynomials symbolic exponents. We present a natural algebraic structure in which such polynomials may be defined and a notion of factorization under which these polynomials form a UFD.