Challenges of Symbolic Computation: My Favorite Open Problems

Abstract

To Bobby F. Caviness on the occasion of his 60th birthdayThe 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.

Approximate factorization of multivariate polynomials via differential equations

“…Example 15 and 15a are polynomials in three variables; 15a is from [15]. Our algorithm employed the method described in Remark 3.4.…”

“…No polynomial-time algorithm is known for computing the nearest factorizable polynomial f [min] , which is open problem 1 in [15]. In [12] a polynomial-time algorithm is given for computing the nearest polynomial with a complex factor of constant degree.…”

Regularization and Matrix Computation in Numerical Polynomial Algebra

“…Computing the approximate irreducible factorization of a multivariate polynomial is the first problem listed in "challenges in symbolic computation" [44] by Kaltofen. The first numerical algorithm with an implementation is developed by Sommese, Verschelde and Wampler [78,79,80].…”

Continuations and monodromy on random riemann surfaces

“…[3], [15,16] or [5] and their bibliography. The reader can also consider [18] for an history of early algorithms. [1] was the first algorithmic paper using monodromy group action as developed below.…”