Resultant properties of gcd of many polynomials and a factorization representation of gcd

Abstract

The properties of the greatest common divisor (gcd) of a set of polynomials of may be investigated using the Sylvester resultant. New properties of the Sylvester resultant linked to gcd are established and these lead to canonical factorizations of resultants expressing the extraction of common divisors from the elements of the original set. These results lead to a new representation of the gcd introduced in terms of a canonical factorization of the Sylvester resultant into a reduced Sylvester resultant and a Toeplitz matrix representing the gcd. The use of the Sylvester resultant allows a simplification of the ERES and matrix pencils computational procedures for gcd computation and provides the means for formulating simpler and robust computational procedures.