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 [math]\displaystyle{ p_0 }[/math] and [math]\displaystyle{ p_1 }[/math] two univariate polynomials. Suppose that they do not have a common root and the degree of [math]\displaystyle{ p_0 }[/math] is greater than the degree of [math]\displaystyle{ p_1 }[/math]. The Sturm series is constructed by:

[math]\displaystyle{ p_i := p_{i+1} q_{i+1} - p_{i+2} \text{ for } i \geq 0. }[/math]

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

Sturm series associated to a characteristic polynomial

Let us see now Sturm series [math]\displaystyle{ p_0,p_1,\dots,p_k }[/math] associated to a characteristic polynomial [math]\displaystyle{ P }[/math] in the variable [math]\displaystyle{ \lambda }[/math]:

[math]\displaystyle{ P(\lambda)= a_0 \lambda^k + a_1 \lambda^{k-1} + \cdots + a_{k-1} \lambda + a_k }[/math]

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

[math]\displaystyle{ \begin{align} 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{align} }[/math]

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

[math]\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 }[/math]

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

[math]\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}. }[/math]

If [math]\displaystyle{ c_{i,0}=0 }[/math] for some [math]\displaystyle{ i }[/math], the quotient [math]\displaystyle{ q_i }[/math] is a higher degree polynomial and the sequence [math]\displaystyle{ p_i }[/math] stops at [math]\displaystyle{ p_h }[/math] with [math]\displaystyle{ h\lt k }[/math].