Sturm series


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

Definition

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

Sturm series associated to a characteristic polynomial

Let us see now Sturm series associated to a characteristic polynomial in the variable :
where for in are rational functions in with the coordinate set. The series begins with two polynomials obtained by dividing by where represents the imaginary unit equal to 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 is equal to which provides the condition. Moreover, the polynomial replaced in the above relation gives the following recursive formulas for computation of the coefficients.
If for some, the quotient is a higher degree polynomial and the sequence stops at with.