Sturm and Sylvester algorithms revisited via tridiagonal determinantal representations

There are several methods to count the number of real roots of an univariate polynomial p(x) ∈ R[x] of degree n (for details we refer to [BPR]). Among them, the Sturm algorithm says that the number of real roots of p(x) is equal to the number of Permanence minus the number of variations of signs which appears in the leading coefficients of the signed remainders sequence of p(x) and p(x). Another method is the Sylvester algorithm which says that the number of real roots of p(x) is equal to the signature of the symmetric matrix whose (i, j)-th entry is the i + j-th Newton sums of the roots of the polynomial p(x). One purpose of the paper is to point out, at least in the generic situation, that these two classical algorithms can be viewed as dual.