Challenges of Symbolic Computation: My Favorite Open Problems

AbstractThe success of the symbolic mathematical computation discipline is striking. The theoretical advances have been continuous and significant: Gröbner bases, the Risch integration algorithm, integer lattice basis reduction, hypergeometric summation algorithms, etc. From the beginning in the early 1960s, it has been the tradition of our discipline to create software that makes our ideas readily available to scientists, engineers, and educators: SAC-1, Reduce, Macsyma, etc. The commercial viability of our system products is proven by Maple and Mathematica. Today’s user communities of symbolic computation systems are diverse: educators, engineers, stock market analysts, etc. The mathematics and computer science in the design and implementation of our algorithms are sophisticated. The research challenges in symbolic computation at the close of the twentieth century are formidable. I state my favorite eight open problems in symbolic computation. They range from problems in symbolic/numeric computing, symbolic algorithm synthesis, to system component construction. I have worked on seven of my problems and borrowed one from George Collins. I present background to each of my problems and a clear-cut test that evaluates whether a proposed attack has solved one of my problems. An additional ninth open problem by Rob Corless and David Jeffrey on complex function semantics is given in an appendix