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:
![{\displaystyle p_{i}:=p_{i+1}q_{i+1}-p_{i+2}{\text{ for }}i\geq 0.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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
:
![{\displaystyle P(\lambda )=a_{0}\lambda ^{k}+a_{1}\lambda ^{k-1}+\cdots +a_{k-1}\lambda +a_{k}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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:
![{\displaystyle {\begin{aligned}p_{0}(\mu )&:=\Re \left({\frac {P(\imath \mu )}{\imath ^{k}}}\right)=a_{0}\mu ^{k}-a_{2}\mu ^{k-2}+a_{4}\mu ^{k-4}\pm \cdots \\p_{1}(\mu )&:=-\Im \left({\frac {P(\imath \mu )}{\imath ^{k}}}\right)=a_{1}\mu ^{k-1}-a_{3}\mu ^{k-3}+a_{5}\mu ^{k-5}\pm \cdots \end{aligned}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
The remaining terms are defined with the above relation. Due to the special structure of these polynomials, they can be written in the form:
![{\displaystyle p_{i}(\mu )=c_{i,0}\mu ^{k-i}+c_{i,1}\mu ^{k-i-2}+c_{i,2}\mu ^{k-i-4}+\cdots }](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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
.
![{\displaystyle c_{i+1,j}=c_{i,j+1}{\frac {c_{i-1,0}}{c_{i,0}}}-c_{i-1,j+1}={\frac {1}{c_{i,0}}}\det {\begin{pmatrix}c_{i-1,0}&c_{i-1,j+1}\\c_{i,0}&c_{i,j+1}\end{pmatrix}}.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
If
for some
, the quotient
is a higher degree polynomial and the sequence
stops at
with
.
References
- ^ (in French) C. F. Sturm. Résolution des équations algébriques. Bulletin de Férussac. 11:419–425. 1829.