Sturm`s Theorem and the Application

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Chapter 1
Sturm’s Theorem
1.1 Sturm’s Thorem
Definition 1.1 We define that the number of changed sign V (a1, &middot; &middot; &middot; , an) for
a = 1, &middot; &middot; &middot; , an ∈ R is as follows.
For the sequense (a1, &middot; &middot; &middot; , an), if ak = 0 (k = 1, &middot; &middot; &middot; , n), we remove ak, and
the removed sequense is redefined (a1, &middot; &middot; &middot; , am). Then we define that the number
of following elements
{i | aiai+1 < 0, 1 &#8804; i <m}
is V (a1, &middot; &middot; &middot; , an). That is to say we look the sign of a1, &middot; &middot; &middot; , am from left to right,
we decide the total number of changed sign is V (a1, &middot; &middot; &middot; , an).

For instance, V (2,&#8722;3, 0,&#8722;1, 2, 0,&#8722;4) = 3.
We apply Euclidian algorithm to f and f = df
dX
for f(X) ∈ R[X].
Now, we define that f(X) ∈ R[X], f0, &middot; &middot; &middot; , fl+1 ∈ R[X] as follows.
First, let f0 = f, f1 = f. Let a quotient of f0 divided f1 is q1, and the
remainder multiplied by &#8722;1 is f2. Let a quotient of f1 divided f2 is q2, and the
remainder multiplied by &#8722;1 is f3. If we continue this operation, the degree of
f0, f1, f2, &middot; &middot; &middot; continue to decline in order, so we get fl = 0 and fl+1 = 0 where
any number l. That is to say
f0 = f
f1 = f = d
dX
f
fi&#8722;1 = qifi &#8722; fi+1 (i = 1, &middot; &middot; &middot; , l)
fl = 0
fl+1 = 0
fl(X) is the greatest common divisor of f(X) and f(X).
Let f = (f0, &middot; &middot; &middot; , fl). f (x) means (f0(x), &middot; &middot; &middot; , fl(x)) for real x.
1
Theorem 1.2 (Sturm’s Theorem) Let a, b ∈ R, a < b, and a and b are not
also multiple root of f, then we can get
V (f (a)) &#8722; V (f (b)) = {x | a < x &#8804; b, f(x) = 0}.


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Sturm’s Theorem and the Application
Katsue Mikami
Hirosaki University
Faculity of Science and Technology
Department of Mathmatical System Science
February. 2006.
Preface
ある多項式が与えられた区間内に実根をいくつ持つかを決定するのが Sturmの定
理である．本論文では第 1 章で Sturmの定理とそれに関連する Fourierの定理や
Descartesの定理に触れ，第 2 章でその応用に触れる．第 2 章では Sturmの定理
を適用することにより一般的な個数を調べたり，解の符号を決定する．さらに解
の間の大小を決定する．
多くの部分で高木貞治著『代数学講義　改定新版』
([1])と Saugata Basu,
Richard Pollack, Marie-Fran¸coise Roy著『Algorithms in Real Algebraic Geom-
etry』([2])に従ったがいくつかの点で補った．
こ..

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