Geodesic
A Basic Result on the Theory of Subresultants
Serdica Journal of Computing, Tome 10 (2016) no. 1, pp. 031-048.

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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.
Keywords: Euclidean Algorithm, Euclidean Polynomial Remainder Sequence (prs), Modified Euclidean prs, Subresultant prs, Modified Subresultant prs, Sylvester Matrices, Bezout Matrix, Sturm’s prs 
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Akritas, Alkiviadis G.; Malaschonok, Gennadi I.; Vigklas, Panagiotis S. A Basic Result on the Theory of Subresultants. Serdica Journal of Computing, Tome 10 (2016) no. 1, pp. 031-048. http://geodesic.mathdoc.fr/item/SJC_2016_10_1_a2/