Institute of Mathematics and Informatics Bulgarian Academy of Sciences
Serdica Journal of Computing, Vol. 10, No 1, (2016), 031p-048p
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.