Sturm series

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In mathematics, the Sturm series^[1] 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:

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

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 $h<k$ .

References

↑ (French) C. F. Sturm. Résolution des équations algébriques. Bulletin de Férussac. 11:419–425. 1829.

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