Polynomials Real Roots Sturm Sequences Sylvester’s Matrices Matrix Triangularization
Institute of Mathematics and Informatics Bulgarian Academy of Sciences
Serdica Journal of Computing, Vol. 7, No 4, (2013), 389p-422p
In 1900 E. B. Van Vleck proposed a very efficient method to compute the Sturm sequence of a polynomial p (x) ∈ Z[x] by triangularizing one of Sylvester’s matrices of p (x) and its derivative p′(x). That method works fine only for the case of complete sequences provided no pivots take place. In 1917, A. J. Pell and R. L. Gordon pointed out this “weakness” in
Van Vleck’s theorem, rectified it but did not extend his method, so that it also works in the cases of: (a) complete Sturm sequences with pivot, and (b) incomplete Sturm sequences.
Despite its importance, the Pell-Gordon Theorem for polynomials in Q[x] has been totally forgotten and, to our knowledge, it is referenced by us for the first time in the literature. In this paper we go over Van Vleck’s theorem and method, modify slightly
the formula of the Pell-Gordon Theorem and present a general triangularization method, called the VanVleck-Pell-Gordon method, that correctly computes in Z[x] polynomial Sturm sequences, both complete and incomplete.