Sturm series
In mathematics, the Sturm series[1] associated with a pair of polynomials is named after Jacques Charles François Sturm.
Definition
Let and
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.
Sturm series associated to a characteristic polynomial
Let us see now Sturm series associated to a characteristic polynomial
in the variable
:
where for
in
are rational functions in
with the coordinate set
. The series begins with two polynomials obtained by dividing
by
where
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 is equal to
which provides the condition
. Moreover, the polynomial
replaced in the above relation gives the following recursive formulas for computation of the coefficients
.
If for some
, the quotient
is a higher degree polynomial and the sequence
stops at
with
.
References
- ↑ (French) C. F. Sturm. Résolution des équations algébriques. Bulletin de Férussac. 11:419–425. 1829.