Abstract: Symbolic Computation in the past 40 years has brought us remarkable theory: Groebner bases, the Risch integration algorithm, polynomial factorization lattice basis reduction, hypergeometric summation algorithms, sparse polynomial interpolation, etc. And it produced remarkable software like Mathematica and Maple, which supplies implementations of these algorithms to the masses. Several system designers even became rich by selling their software. As it turned out, a killer application of computer algebra is high energy physics, where a special purpose computer algebra system, SCHOONSHIP, helped in work worthy of a Nobel Prize in physics in 1999.

In my talk I will give glimpses what the descipline of Symbolic Computation is and what problems it tackles. In particular, I will discuss the use of heuristics (numerical, randomized, and algorithmic) that seem necessary to solve some of today's problems in geometric modeling and equation solving. Thus, we seem to have come full cycle (the discipline may have started in the 1960s at the MIT AI Lab), but with a twist that I shall explain.

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