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:
:$p_i := p_ q_ - p_ \text i \geq 0.$
This is almost the same algorithm as Euclid's but the remainder $p_$ 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 $\lambda$:
:$P(\lambda)= a_0 \lambda^k + a_1 \lambda^ + \cdots + a_ \lambda + a_k$
where $a_i$ for $i$ in $\$ are rational functions in $\mathbb(Z)$ 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 $\sqrt$ and separate real and imaginary parts:
:$\begin p_0(\mu) & := \Re \left(\frac\right ) = a_0 \mu^k - a_2 \mu^ + a_4 \mu^ \pm \cdots \\ p_1(\mu) & := -\Im \left( \frac\right)= a_1 \mu^ - a_3 \mu^ + a_5 \mu^ \pm \cdots \end$

The remaining terms are defined with the above relation. Due to the special structure of these polynomials, they can be written in the form:
:$p_i(\mu)= c_ \mu^ + c_ \mu^ + c_ \mu^+\cdots$
In these notations, the quotient $q_i$ is equal to $(c_/c_)\mu$ which provides the condition $c_\neq 0$. Moreover, the polynomial $p_i$ replaced in the above relation gives the following recursive formulas for computation of the coefficients $c_$.
:$c_= c_ \frac-c_ = \frac \det \begin c_ & c_ \\ c_ & c_ \end.$
If $c_=0$ for some $i$, the quotient $q_i$ is a higher degree polynomial and the sequence $p_i$ stops at $p_h$ with h.

References

{{Reflist, refs=
{{in lang, fr C. F. Sturm. Résolution des équations algébriques. Bulletin de Férussac. 11:419–425. 1829.

Mathematical series