In mathematics, the Sturm series associated with a pair of polynomials is named after Jacques Charles François Sturm.

Definition

Let p_0 and p_1 two univariate polynomials. Suppose that they do not have a common root and the degree of p_0 is greater than the degree of p_1. The Sturm series is constructed by: This is almost the same algorithm as Euclid's but the remainder p_{i+2} has negative sign.

Sturm series associated to a characteristic polynomial

Let us see now Sturm series associated to a characteristic polynomial P in the variable \lambda: where a_i for i in are rational functions in with the coordinate set Z. The series begins with two polynomials obtained by dividing by \imath ^k where \imath represents the imaginary unit equal to \sqrt{-1} and separate real and imaginary parts: The remaining terms are defined with the above relation. Due to the special structure of these polynomials, they can be written in the form: In these notations, the quotient q_i is equal to which provides the condition. Moreover, the polynomial p_i replaced in the above relation gives the following recursive formulas for computation of the coefficients c_{i,j}. If c_{i,0}=0 for some i, the quotient q_i is a higher degree polynomial and the sequence p_i stops at p_h with h<k.