Three New Methods for Computing Subresultant Polynomial Remainder Sequences (PRS’S)
Alkiviadis G. Akritas · Serdica Journal of Computing · 2015
Given the polynomials f, g ∈ Z[x] of degrees n, m, respectively,with n > m, three new, and easy to understand methods — along withthe more efficient variants of the last two of them — are presented for thecomputation of their subresultant polynomial remainder sequence (prs).All three methods evaluate a single determinant (subresultant) of anappropriate sub-matrix of sylvester1, Sylvester’s widely known and usedmatrix of 1840 of dimension (m + n) × (m + n), in order to compute thecorrect sign of each polynomial in the sequence and — except for the secondmethod — 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 operationsin Q[x], whereas the third one triangularizes sylvester2, Sylvester’s littleknown 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