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Title: A Basic Result on the Theory of Subresultants
Authors: Akritas, Alkiviadis G.
Malaschonok, Gennadi I.
Vigklas, Panagiotis S.
Keywords: Euclidean Algorithm
Euclidean Polynomial Remainder Sequence (prs)
Modified Euclidean prs
Subresultant prs
Modified Subresultant prs
Sylvester Matrices
Bezout Matrix
Sturm’s prs
Issue Date: 2016
Publisher: Institute of Mathematics and Informatics Bulgarian Academy of Sciences
Citation: Serdica Journal of Computing, Vol. 10, No 1, (2016), 031p-048p
Abstract: Given the polynomials f, g ∈ Z[x] the main result of our paper, Theorem 1, establishes a direct one-to-one correspondence between the modified Euclidean and Euclidean polynomial remainder sequences (prs’s) of f, g computed in Q[x], on one hand, and the subresultant prs of f, g computed by determinant evaluations in Z[x], on the other. An important consequence of our theorem is that the signs of Euclidean and modified Euclidean prs’s - computed either in Q[x] or in Z[x] - are uniquely determined by the corresponding signs of the subresultant prs’s. In this respect, all prs’s are uniquely "signed". Our result fills a gap in the theory of subresultant prs’s. In order to place Theorem 1 into its correct historical perspective we present a brief historical review of the subject and hint at certain aspects that need - according to our opinion - to be revised. ACM Computing Classification System (1998): F.2.1, G.1.5, I.1.2.
ISSN: 1312-6555
Appears in Collections:Volume 10 Number 1

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