Sturm Sequences and Modified Subresultant Polynomial Remainder Sequences

Abstract

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.