Challenges of Symbolic Computation - My Favorite Open Problems
The success of the symbolic mathematical computation discipline is striking. The theoretical advances have been continuous and significant: Groebner, Risch, LLL, WZ, etc. From the beginning in the early 60s, 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 20th century are formidable.
I state my favorite nine open problems in symbolic computation. They range from problems in symbolic/numeric computing, symbolic algorithm synthesis, system component construction, to changing the opinion of Donald Knuth. I have worked on eight 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.
Erich Kaltofen
Mathematics Department
North Carolina State University
kaltofen@math.ncsu.edu