In mathematics, the Sturm series[1] associated with a pair of polynomials is named after Jacques Charles François Sturm.
two univariate polynomials.
Suppose that they do not have a common root and the degree of
is greater than the degree of
The Sturm series is constructed by: This is almost the same algorithm as Euclid's but the remainder
has negative sign.
Let us see now Sturm series
associated to a characteristic polynomial
in the variable
λ
are rational functions in
with the coordinate set
The series begins with two polynomials obtained by dividing
( ı μ )
ı
represents the imaginary unit equal to
and separate real and imaginary parts: The remaining terms are defined with the above relation.
Due to the special structure of these polynomials, they can be written in the form: In these notations, the quotient
) μ
which provides the condition
Moreover, the polynomial
replaced in the above relation gives the following recursive formulas for computation of the coefficients
, the quotient
is a higher degree polynomial and the sequence