Sturm theorem
If
![]() | (*) |
is a Sturm series on the interval ,
, and
is the number of variations of sign in the series (*) at a point
(vanishing terms are not taken into consideration), then the number of distinct roots of the function
on the interval
is equal to the difference
.
A Sturm series (or Sturm sequence) is a sequence of real-valued continuous functions (*) on having a finite number of roots on this interval, and such that
1) ;
2) on
;
3) from for some
and given
in
it follows that
;
4) from for a given
it follows that for sufficiently small
,
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This theorem was proved by J.Ch. Sturm [1], who also proposed the following method of constructing a Sturm series for a polynomial with real coefficients and without multiple roots:
,
, and, if the polynomials
are already constructed, then as
one should take minus the remainder occurring in the process of dividing
by
. Here,
will be a non-zero constant.
References
[1] | J.Ch. Sturm, Bull. de Férussac , 11 (1829) |
[2] | A.G. Kurosh, "Higher algebra" , MIR (1972) (Translated from Russian) |
Comments
The coefficients of the polynomials in the Sturm series must belong to a real-closed field. The algorithm to determine a Sturm series for a polynomial can be described as follows:
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so is a non-zero constant.
References
[a1] | N. Jacobson, "Basic algebra" , I , Freeman (1974) |
[a2] | L.E.J. Dickson, "New first course in the theory of equations" , Wiley (1939) |
[a3] | B.L. van der Waerden, "Algebra" , 1 , Springer (1967) (Translated from German) |
Sturm theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sturm_theorem&oldid=48887