Sturm series

Let ${\displaystyle p_{0}}$ and ${\displaystyle p_{1}}$ two univariate polynomials. Suppose that they do not have a common root and the degree of ${\displaystyle p_{0}}$ is greater than the degree of ${\displaystyle p_{1}}$. The Sturm series is constructed by:

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

This is almost the same algorithm as Euclid's but the remainder ${\displaystyle p_{i+2}}$ has negative sign.

Let us see now Sturm series ${\displaystyle p_{0},p_{1},\dots ,p_{k}}$ associated to a characteristic polynomial ${\displaystyle P}$ in the variable ${\displaystyle \lambda }$:

${\displaystyle P(\lambda )=a_{0}\lambda ^{k}+a_{1}\lambda ^{k-1}+\cdots +a_{k-1}\lambda +a_{k}}$

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

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

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

${\displaystyle p_{i}(\mu )=c_{i,0}\mu ^{k-i}+c_{i,1}\mu ^{k-i-2}+c_{i,2}\mu ^{k-i-4}+\cdots }$

In these notations, the quotient ${\displaystyle q_{i}}$ is equal to ${\displaystyle (c_{i-1,0}/c_{i,0})\mu }$ which provides the condition ${\displaystyle c_{i,0}\neq 0}$. Moreover, the polynomial ${\displaystyle p_{i}}$ replaced in the above relation gives the following recursive formulas for computation of the coefficients ${\displaystyle c_{i,j}}$.

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

If ${\displaystyle c_{i,0}=0}$ for some ${\displaystyle i}$, the quotient ${\displaystyle q_{i}}$ is a higher degree polynomial and the sequence ${\displaystyle p_{i}}$ stops at ${\displaystyle p_{h}}$ with ${\displaystyle h<k}$.