In mathematics, the Sturm series associated with a pair of polynomials is named after Jacques Charles François Sturm.
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 ≥ 0. This is almost the same algorithm as Euclid's but the remainder p i + 2 has negative sign.
Let us see now Sturm series p 0 , p 1 , … , p k associated to a characteristic polynomial P in the variable λ :
P ( λ ) = a 0 λ k + a 1 λ k − 1 + ⋯ + a k − 1 λ + a k where a i for i in { 1 , … , 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 − 1 and separate real and imaginary parts:
p 0 ( μ ) := ℜ ( P ( ı μ ) ı k ) = a 0 μ k − a 2 μ k − 2 + a 4 μ k − 4 ± ⋯ p 1 ( μ ) := − ℑ ( P ( ı μ ) ı k ) = a 1 μ k − 1 − a 3 μ k − 3 + a 5 μ k − 5 ± ⋯ 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 + ⋯ 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 ≠ 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 c i − 1 , 0 c i , 0 − c i − 1 , j + 1 = 1 c i , 0 det ( c i − 1 , 0 c i − 1 , j + 1 c i , 0 c i , j + 1 ) . 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 .
