Gauss: A Parameterized Domain of Computation System with Support for Signature Functions

A b s t r a c t . The fastest known algorithms in classical algebra make use of signature functions. That is, reducing computat ion with formulae to computing with the integers modulo p, by substi tut ing random numbers for variables, and mapping constants modulo p. This idea is exploited in specific algorithms in computer algebra systems, e.g. algorithms for polynomial greatest common divisors. It is also used as a heuristic to speed up other calculations. But none exploit it in a systematic manner. The goal of this work was twofold. First, to design an AXIOM like system in which these signature functions can be constructed automatically, hence bet ter exploited, and secondly, to exploit them in new ways. In this paper we report on the design of such a system, Gauss.