On a Theorem by Van Vleck Regarding Sturm Sequences

Abstract

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.