In mathematics, the Sturm series associated with a pair of polynomials is named after Jacques Charles François Sturm.
Definition
Let p 0 {\displaystyle p_{0}} and p 1 {\displaystyle p_{1}} two univariate polynomials. Suppose that they do not have a common root and the degree of p 0 {\displaystyle p_{0}} is greater than the degree of p 1 {\displaystyle p_{1}} . The Sturm series is constructed by:
p i := p i + 1 q i + 1 − p i + 2 for i ≥ 0. {\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 p i + 2 {\displaystyle p_{i+2}} has negative sign.
Sturm series associated to a characteristic polynomial Let us see now Sturm series p 0 , p 1 , … , p k {\displaystyle p_{0},p_{1},\dots ,p_{k}} associated to a characteristic polynomial P {\displaystyle P} in the variable λ {\displaystyle \lambda } :
P ( λ ) = a 0 λ k + a 1 λ k − 1 + ⋯ + a k − 1 λ + a k {\displaystyle P(\lambda )=a_{0}\lambda ^{k}+a_{1}\lambda ^{k-1}+\cdots +a_{k-1}\lambda +a_{k}}
where a i {\displaystyle a_{i}} for i {\displaystyle i} in { 1 , … , k } {\displaystyle \{1,\dots ,k\}} are rational functions in R ( Z ) {\displaystyle \mathbb {R} (Z)} with the coordinate set Z {\displaystyle Z} . The series begins with two polynomials obtained by dividing P ( ı μ ) {\displaystyle P(\imath \mu )} by ı k {\displaystyle \imath ^{k}} where ı {\displaystyle \imath } represents the imaginary unit equal to − 1 {\displaystyle {\sqrt {-1}}} and separate real and imaginary parts:
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