Subresultants and Reduced Polynomial Remainder Sequences

Abstract

Let @@@@ be an integral domain, P(@@@@) the integral domain of polynomials over @@@@. Let P , Q ∈ P(@@@@) with m @@@@ deg ( P ) ≥ n = deg ( Q ) > 0. Let M be the matrix whose determinant defines the resultant of P and Q . Let M ij be the submatrix of M obtained by deleting the last j rows of P coefficients, the last j rows of Q coefficients and the last 2 j +1 columns, excepting column m — n — i — j (0 ≤ i ≤ j < n ). The polynomial R j ( x ) = ∑ i i =0 det ( M ij ) x i is the j-t subresultant of P and Q , R 0 being the resultant. If b = £( Q ), the leading coefficient of Q , then exist uniquely R , S ∈ P(@@@@) such that b m-n +1 P = QS + R with deg ( R ) < n ; define R( P , Q ) = R . Define P i ∈ P( F ), F the quotient field of @@@@, inductively: P 1 = P , P 2 = Q , P 3 = R P 1 , P 2 P i -2 = R( P i , P i +1 )/ c δ i -1 +1 i for i ≥ 2 and n i +1 > 0, where c i = £( P i ), n i = deg ( P i ) and δ i = n i — n i +1 . P 1 , P 2 , …, P k , for k ≥ 3, is called a reduced polynomial remainder sequence . Some of the main results are: (1) P i ∈ P(@@@@) for 1 ≤ i ≤ k ; (2) P k = ± A k R n k -1 -1 , when A k = Π k -2 i -2 c δ i -1 (δ i -1) i ; (3) c δ k -1 -1 k P k = ± A k +1 R n k ; (4) R j = 0 for n k < j < n k -1 — 1. Taking @@@@ to be the integers I , or P r ( I ), these results provide new algorithms for computing resultant or greatest common divisors of univariate or multivariate polynomials. Theoretical analysis and extensive testing on a high-speed computer show the new g.c.d. algorithm to be faster than known algorithms by a large factor. When applied to bivariate polynomials, for example this factor grows rapidly with the degree and exceeds 100 in practical cases.