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Title: On the Remainders Obtained in Finding the Greatest Common Divisor of Two Polynomials
Authors: Akritas, Alkiviadis
Malaschonok, Gennadi
Vigklas, Panagiotis
Keywords: Polynomial Remainder Sequence (PRS)
Sylvester’s Matrices
Euclidean PRS
Subresultant PRS
Sturm Sequence
Modified Subresultant PRS
Issue Date: 2015
Publisher: Institute of Mathematics and Informatics Bulgarian Academy of Sciences
Citation: Serdica Journal of Computing, Vol. 9, No 2, (2015), 123p-138p
Abstract: In 1917 Pell (1) and Gordon used sylvester2, Sylvester’s little known and hardly ever used matrix of 1853, to compute(2) the coefficients of a Sturmian remainder — obtained in applying in Q[x], Sturm’s algorithm on two polynomials f, g ∈ Z[x] of degree n — in terms of the determinants (3) of the corresponding submatrices of sylvester2. Thus, they solved a problem that had eluded both J. J. Sylvester, in 1853, and E. B. Van Vleck, in 1900. (4) In this paper we extend the work by Pell and Gordon and show how to compute (2) the coefficients of an Euclidean remainder — obtained in finding in Q[x], the greatest common divisor of f, g ∈ Z[x] of degree n — in terms of the determinants (5) of the corresponding submatrices of sylvester1, Sylvester’s widely known and used matrix of 1840. (1) See the link for her biography (2) Both for complete and incomplete sequences, as defined in the sequel. (3) Also known as modified subresultants. (4) Using determinants Sylvester and Van Vleck were able to compute the coefficients of Sturmian remainders only for the case of complete sequences. (5) Also known as (proper) subresultants.
ISSN: 1312-6555
Appears in Collections:Volume 9 Number 2

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