Polynomial remainder sequence and approximate GCD

Abstract

Let P 1 and P 2 be polynomials, univariate or multivariate, and let (P 1 , P 2 , P 3 ,…, P i ,…) be a polynomial remainder sequence. Let A i and B i (i = 3, 4,…) be polynomials such that A i P 1 + B i P 2 = P i , deg(A i ) < deg(P 2 ) - deg(P i ), deg(B i ) < deg(P 1 ) - deg(P i ), where the degree is for the main variable. We derive relations such as C i P 1 = -B i+1 P i + B i P i+1 and C i P 2 = A i+1 P i - A i P i+1 , where C i is independent of the main variable. Using these relations, we discuss approximate common divisors calculated by polynomial remainder sequence.