On Degrees of Modular Common Divisors and the Big PrimegcdAlgorithm

Abstract

We consider a few modifications of the Big prime modulargcdalgorithm for polynomials inZ[x]. Our modifications are based on bounds of degrees of modular common divisors of polynomials, on estimates of the number of prime divisors of a resultant, and on finding preliminary bounds on degrees of common divisors using auxiliary primes. These modifications are used to suggest improved algorithms forgcdcalculation and for coprime polynomials detection. To illustrate the ideas we apply the constructed algorithms on certain polynomials, in particular on polynomials from Knuth’s example of intermediate expression swell.