Please use this identifier to cite or link to this item: http://hdl.handle.net/10525/2487

 Title: On the Remainders Obtained in Finding the Greatest Common Divisor of Two Polynomials Authors: Akritas, AlkiviadisMalaschonok, GennadiVigklas, Panagiotis Keywords: Polynomial Remainder Sequence (PRS)Sylvester’s MatricesEuclidean PRSSubresultant PRSSturm SequenceModified 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 http://en.wikipedia.org/wiki/Anna_Johnson_Pell_Wheeler 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. URI: http://hdl.handle.net/10525/2487 ISSN: 1312-6555 Appears in Collections: Volume 9 Number 2

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