Various New Methods for Computing Subresultant Polynomial Remainder Sequences (PRS’s)

Teaching subresultant prs’s is an unpleasant experience because there is a misunderstanding about the role of Sylvester’s two matrices and how they affect the signs of the sequences. Almost all articles and texts on the subject perform operations in Z[x] and use a form of pseudo-division that distorts the signs of the polynomial remainders; hence, sentences like “forget about the signs” appear quite often in the literature. In this talk we clarify the mystery about the signs and show how to compute the subresultant prs’s in various ways — performing operations even in Q[x]. Briefly stated, here is how. Consider the polynomials f ,g ∈ Z[x] of degrees n,m, respectively, with n > m. We call Euclidean prs the sequence of polynomial remainders obtained during the execution of the Euclidean algorithm for polynomial gcd. If the polynomials in the sequence are of degrees n = m+ 1,m,m− 1,m− 2, . . . ,0, the sequence is called complete. Otherwise it is called incomplete. The complete Euclidean prs of two polynomials can be computed either by doing polynomial divisions over the integers/rationals or by evaluating determinants of submatrices of sylvester1 — J.J. Sylvester’s matrix of 1840 [1], [12]. In the latter case, the coefficients of each polynomial remainder are the above mentioned determinants (or subresultants) and we are talking about the subresultant prs [6], [7], [8], [10]. Caveat 1: As demonstrated by f = x8 + x6−3x4−3x3 +8x2 +2x−5 and g = 3x6+5x4−4x2−9x+21, the signs of the polynomials in an incomplete Euclidean prs may differ from those of the corresponding subresultant prs [3]. Analogous to Euclidean prs’s are the Sturm sequences, which are obtained by modifying Euclid’s algorithm; that is, at each step we take the negative of the remainder obtained. Like their cousins, complete Sturm sequences can be computed either by doing polynomial divisions over the integers/rationals or by evaluating determinants of