Subresultants and Reduced Polynomial Remainder Sequences

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 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 provi