Sturm series

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:

p_i := p_{i+1} q_{i+1} - p_{i+2} \text{ for } i \geq 0. $$ This is almost the same algorithm as Euclid's but the remainder $$p_{i+2}$$ has negative sign.

Sturm series associated to a characteristic polynomial
Let us see now Sturm series $$p_0,p_1,\dots,p_k$$ associated to a characteristic polynomial $$P$$ in the variable $$\lambda$$:

P(\lambda)= a_0 \lambda^k + a_1 \lambda^{k-1} + \cdots + a_{k-1} \lambda + a_k $$ where $$a_i$$ for $$i$$ in $$\{1,\dots,k\}$$ are rational functions in $$\mathbb{R}(Z)$$ with the coordinate set $$Z$$. The series begins with two polynomials obtained by dividing $$P(\imath \mu)$$ by $$\imath ^k$$ where $$\imath$$ represents the imaginary unit equal to $$\sqrt{-1}$$ and separate real and imaginary parts:

\begin{align} p_0(\mu) & := \Re \left(\frac{P(\imath \mu)}{\imath^k}\right ) = a_0 \mu^k - a_2 \mu^{k-2} + a_4 \mu^{k-4} \pm \cdots \\ p_1(\mu) & := -\Im \left( \frac{P(\imath \mu)}{\imath^k}\right)= a_1 \mu^{k-1} - a_3 \mu^{k-3} + a_5 \mu^{k-5} \pm \cdots \end{align} $$

The remaining terms are defined with the above relation. Due to the special structure of these polynomials, they can be written in the form:

p_i(\mu)= c_{i,0} \mu^{k-i} + c_{i,1} \mu^{k-i-2} + c_{i,2} \mu^{k-i-4}+\cdots $$ In these notations, the quotient $$q_i$$ is equal to $$(c_{i-1,0}/c_{i,0})\mu$$ which provides the condition $$c_{i,0}\neq 0$$. Moreover, the polynomial $$p_i$$ replaced in the above relation gives the following recursive formulas for computation of the coefficients $$c_{i,j}$$.

c_{i+1,j}= c_{i,j+1} \frac{c_{i-1,0}}{c_{i,0}}-c_{i-1,j+1} = \frac{1}{c_{i,0}} \det \begin{pmatrix} c_{i-1,0} & c_{i-1,j+1} \\ c_{i,0} & c_{i,j+1} \end{pmatrix}. $$ If $$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$$.