Sturm series

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

Definition

Further information: Sturm chain

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:

$p_i:=p_{i+1}{q}_{i+1}-p_{i+2}\text{ for }i\ge 0.$

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 $P(ı\mu )$ by ${ı}^k$ where $ı$ represents the imaginary unit equal to $\sqrt{-1}$ and separate real and imaginary parts:

$\begin{array}{rl}{p}_0(\mu )& :=\mathrm{\Re }\left(\frac{P(ı\mu )}{{ı}^k}\right)=a_0{\mu }^k-a_2{\mu }^{k-2}+a_4{\mu }^{k-4}\pm \cdots \\ p_1(\mu )& :=-\mathrm{\Im }\left(\frac{P(ı\mu )}{{ı}^k}\right)=a_1{\mu }^{k-1}-a_3{\mu }^{k-3}+a_5{\mu }^{k-5}\pm \cdots \end{array}$

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 $c_{i,0}\ne 0$. Moreover, the polynomial $p_i$ replaced in the above relation gives the following recursive formulas for computation of the coefficients $c_{i,j}$.

$c_{i+1,j}=c_{i,j+1}\frac{c_{i-1,0}}{c_{i,0}}-c_{i-1,j+1}=\frac{1}{c_{i,0}}\det \left(\begin{array}{cc}{c}_{i-1,0}& c_{i-1,j+1}\\ c_{i,0}& c_{i,j+1}\end{array}\right).$

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 $h<k$.