A Quick Fail-Safe Procedure for Determining Whether the GCD of 2 Polynomials is 1

Abstract

One of the most widely used routines in an algebraic manipulation system is a polynomial manipulation package (1,2,3). The crucial operation in such routines is the extraction of the Greatest Common Divisor (GCD) of two polynomials. This operation is crucial because of its frequent use and because it is an expensive operation in regard to time and space. Experiments by Collins(1) have shown that given two polynomials chosen at random, the GCD has a high probability of being 1. Taking into account this probability and the cost of obtaining a GCD (some GCDs of polynomials of degree 5 in two or three variables can take on the order of a minute on the 7094(1), it appears that a quick method of determining whether the GCD is exactly 1 would be profitable. While no such complete method is known to exist, a fail-safe procedure has been found and is described here. A fail-safe procedure is characterized by the fact that when it comes to decision (in this case that the GCD is 1), then the decision is correct. However, the conclusion (i.e. that the GCD is 1) may be true, and the procedure need not arrive at a decision regarding it. It is believed that the fail-safe procedure presented here (and its extension to the linear case) will arrive at a decision quite frequently when the GCD is actually 1.