Sturm and Sylvester algorithms revisited via tridiagonal determinantal representations

Ronan Quarez (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

First, we show that Sturm algorithm and Sylvester algorithm, which compute the number of real roots of a given univariate polynomial, lead to two dual tridiagonal determinantal representations of the polynomial. Next, we show that the number of real roots of a polynomial given by a tridiagonal determinantal representation is greater than the signature of this representation.

Mots clés: Determinantal representation; Hankel matrix; LU decomposition; Newton sums; Real roots; Signed remainders sequence; Sturm algorithm; Sylvester algorithm; Tridiagonal matrix

Domaines: Anneaux et algèbres [math.RA]

Ronan Quarez. Sturm and Sylvester algorithms revisited via tridiagonal determinantal representations. Linear Algebra and its Applications, 2010, 433 (6), pp.1082-1100. ⟨10.1016/j.laa.2010.04.049⟩. ⟨hal-00338925⟩