1924–25 Beşiktaş J.K. season

In mathematics, the Sturm series^[1] associated with a pair of polynomials is named after Jacques Charles François Sturm.

Definition

Template:See 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} 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 $λ$ :

P (λ) = a_{0} λ^{k} + a_{1} λ^{k - 1} + \dots + a_{k - 1} λ + a_{k}

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

\begin{matrix} p_{0} (μ) & := ℜ (\frac{P (ı μ)}{ı^{k}}) = a_{0} μ^{k} - a_{2} μ^{k - 2} + a_{4} μ^{k - 4} \pm \dots \\ p_{1} (μ) & := - ℑ (\frac{P (ı μ)}{ı^{k}}) = a_{1} μ^{k - 1} - a_{3} μ^{k - 3} + a_{5} μ^{k - 5} \pm \dots \end{matrix}

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} (μ) = c_{i, 0} μ^{k - i} + c_{i, 1} μ^{k - i - 2} + c_{i, 2} μ^{k - i - 4} + \dots

In these notations, the quotient $q_{i}$ is equal to $(c_{i - 1, 0} / c_{i, 0}) μ$ 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{matrix} c_{i - 1, 0} & c_{i - 1, j + 1} \\ c_{i, 0} & c_{i, j + 1} \end{matrix}) .

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

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[1]

1924–25 Beşiktaş J.K. season

Definition

Sturm series associated to a characteristic polynomial

References

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