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:

$p_{i}:=p_{i+1}q_{i+1}-p_{i+2}{\text{ for }}i\geq 0.$

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:

${\begin{aligned}p_{0}(\mu )&:=\Re \left({\frac {P(\imath \mu )}{\imath ^{k}}}\right)=a_{0}\mu ^{k}-a_{2}\mu ^{k-2}+a_{4}\mu ^{k-4}\pm \cdots \\p_{1}(\mu )&:=-\Im \left({\frac {P(\imath \mu )}{\imath ^{k}}}\right)=a_{1}\mu ^{k-1}-a_{3}\mu ^{k-3}+a_{5}\mu ^{k-5}\pm \cdots \end{aligned}}$

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 $h<k$.