Subresultants and Reduced Polynomial Remainder Sequences

• Authors: George E. Collins
• Venue: Journal of the ACM (JACM)
• Year: 1967

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 Mij 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 2j+1 columns, excepting column m — n — i — j (0 ≤ i ≤ j n). The polynomial Rj(x) = ∑ii=0 det (Mij)xi is the j-t subresultant of P and Q, R0 being the resultant. If b = £(Q), the leading coefficient of Q, then exist uniquely R, S ∈ P(@@@@) such that bm-n+1 P = QS + R with deg (R) n; define R(P, Q) = R. Define Pi ∈ P(F), F the quotient field of @@@@, inductively: P1 = P, P2 = Q, P3 = RP1, P2 Pi-2 = R(Pi, Pi+1)/c&dgr;i-1+1i for i ≥ 2 and ni+1 0, where ci = £(Pi), ni = deg (Pi) and &dgr;i = ni — ni+1. P1, P2, …, Pk, for k ≥ 3, is called a reduced polynomial remainder sequence. Some of the main results are: (1) Pi ∈ P(@@@@) for 1 ≤ i ≤ k; (2) Pk = ± AkRnk-1-1, when Ak = &Pgr;k-2i-2c&dgr;i-1(&dgr;i-1)i; (3) c&dgr;k-1-1k Pk = ±Ak+1Rnk; (4) Rj = 0 for nk j nk-1 — 1. Taking @@@@ to be the integers I, or Pr(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.