Comparing the speed of programs for sparse polynomial multiplication

How should one design and implement a program for the multiplication of sparse polynomials? This is a simple question, necessarily addressed by the builders of any computer algebra system (CAS). To examine a few options we start with a single easily-stated computation which we believe represents a useful benchmark of "medium difficulty" for CAS designs. We describe a number of design options and their effects on performance. We also examine the performance of a variety of commercial and freely-distributed systems. Important considerations include the cost of high-precision (exact) integer arithmetic and the effective use of cache memory.