Three New Methods for Computing Subresultant Polynomial Remainder Sequences (PRS’S)

Abstract

Given the polynomials f, g ∈ Z[x] of degrees n, m, respectively,
with n > m, three new, and easy to understand methods — along with
the more efficient variants of the last two of them — are presented for the
computation of their subresultant polynomial remainder sequence (prs).
All three methods evaluate a single determinant (subresultant) of an
appropriate sub-matrix of sylvester1, Sylvester’s widely known and used
matrix of 1840 of dimension (m + n) × (m + n), in order to compute the
correct sign of each polynomial in the sequence and — except for the second
method — to force its coefficients to become subresultants.
Of interest is the fact that only the first method uses pseudo remainders.

The second method uses regular remainders and performs operations
in Q[x], whereas the third one triangularizes sylvester2, Sylvester’s little
known and hardly ever used matrix of 1853 of dimension 2n × 2n.
All methods mentioned in this paper (along with their supporting functions)
have been implemented in Sympy and can be downloaded from the link
http://inf-server.inf.uth.gr/~akritas/publications/subresultants.py