Polynomials Real Roots Sturm Sequences Sylvester’s Matrices Matrix Triangularization
Institute of Mathematics and Informatics Bulgarian Academy of Sciences
Serdica Journal of Computing, Vol. 8, No 1, (2014), 29p-46p
In 1971 using pseudo-divisions - that is, by working in Z[x] -
Brown and Traub computed Euclid’s polynomial remainder sequences (prs’s)
and (proper) subresultant prs’s using sylvester1, the most widely known
form of Sylvester’s matrix, whose determinant defines the resultant of two
polynomials. In this paper we use, for the first time in the literature, the Pell-Gordon
Theorem of 1917, and sylvester2, a little known form of Sylvester’s matrix of 1853
to initially compute Sturm sequences in Z[x] without pseudodivisions -
that is, by working in Q[x]. We then extend our work in Q[x] and, despite
the fact that the absolute value of the determinant of
sylvester2 equals the absolute value of the resultant, we construct modified
subresultant prs’s, which may differ from the proper ones only in sign.
ACM Computing Classification System (1998): F.2.1, G.1.5, I.1.2.