Subresultants and Reduced Polynomial Remainder Sequences
Subresultants and Reduced Polynomial Remainder Sequences
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DOI：
10.1145/321371.321381
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发表时间：
1967
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通讯作者：
G. Collins
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文献类型：
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作者：
G. Collins
@@@成为一个整体域，P（@@@）通过@@@ @@@Q∈P（@@@）与m @@@@n = N = N = DEG（Q）>0。令M为矩阵，其确定器定义了P和Q的结果。让MIJ是通过删除P Coreifics的最后一个J行，Q Coreifics的最后J行和最后一个获得的M获得的M 2J+1列，除M-n-i - j（0≤i≤Jn）外结果r（p，q）=R。定义pi∈P（f），f @@@@@的商字段，归纳：p1 = p，p2 = q，p3 = rp1，p2 pi-2 = r（pi，pi，pi，pi，pi， I≥2和Ni+1> 0的Pi+1）/CΔI-1+1i，其中Ci =£（pi），ni = deg（pi）和ΔI= ni = ni-ni-ni+1。对于K≥3，PK称为减少的多项式序列。 1，ak =＆pgr; k-2i-2cΔi-1（Δi-1）i；以@@@的方式成为整数，或PR（i），这些结果提供了用于计算结果的新算法，或者是单变量或多个多项式分析的最大常见分隔线。 G.C.D.算法比已知的算法更快，例如，在双变量多项式上，该因素随着程度而迅速增长，在实际情况下超过了100个。
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)/cδi-1+1i for i ≥ 2 and ni+1 > 0, where ci = £(Pi), ni = deg (Pi) and δi = 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-2cδi-1(δi-1)i; (3) cδk-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.