A Basic Result on the Theory of Subresultants

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.