- Definition
- Sturm series associated to a characteristic polynomial
- References
In mathematics, the Sturm series associated with a pair of polynomials is named after Jacques Charles François Sturm.
Definition
{{See|Sturm chain}}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
C. F. Sturm. Résolution des équations algébriques. Bulletin de Férussac. 11:419–425. 1829.}}
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