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If
 
If
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090790/s0907901.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{* }
 +
f _ {0} ( x), \ldots, f _ {s} ( x)
 +
$$
  
is a Sturm series on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090790/s0907902.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090790/s0907903.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090790/s0907904.png" /> is the number of variations of sign in the series (*) at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090790/s0907905.png" /> (vanishing terms are not taken into consideration), then the number of distinct roots of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090790/s0907906.png" /> on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090790/s0907907.png" /> is equal to the difference <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090790/s0907908.png" />.
+
is a Sturm series on the interval $  [ a, b] $,  
 +
$  a < b $,  
 +
and $  w( x) $
 +
is the number of variations of sign in the series (*) at a point $  x \in [ a, b] $
 +
(vanishing terms are not taken into consideration), then the number of distinct roots of the function $  f _ {0} $
 +
on the interval $  [ a, b] $
 +
is equal to the difference $  w( a)- w( b) $.
  
A Sturm series (or Sturm sequence) is a sequence of real-valued continuous functions (*) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090790/s0907909.png" /> having a finite number of roots on this interval, and such that
+
A Sturm series (or Sturm sequence) is a sequence of real-valued continuous functions (*) on $  [ a, b] $
 +
having a finite number of roots on this interval, and such that
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090790/s09079010.png" />;
+
1) $  f _ {0} ( a) f _ {0} ( b) \neq 0 $;
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090790/s09079011.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090790/s09079012.png" />;
+
2) $  f _ {s} ( x) \neq 0 $
 +
on $  [ a, b] $;
  
3) from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090790/s09079013.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090790/s09079014.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090790/s09079015.png" /> and given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090790/s09079016.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090790/s09079017.png" /> it follows that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090790/s09079018.png" />;
+
3) from $  f _ {k} ( c) = 0 $
 +
for some $  k $
 +
$  ( 0 < k < s) $
 +
and given $  c $
 +
in $  [ a, b] $
 +
it follows that $  f _ {k-1} ( c) f _ {k+1} ( c) < 0 $;
  
4) from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090790/s09079019.png" /> for a given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090790/s09079020.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090790/s09079021.png" /> it follows that for sufficiently small <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090790/s09079022.png" />,
+
4) from $  f _ {0} ( c) = 0 $
 +
for a given $  c $
 +
$  ( a < c < b) $
 +
it follows that for sufficiently small $  \epsilon > 0 $,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090790/s09079023.png" /></td> </tr></table>
+
$$
 +
f _ {0} ( x) f _ {1} ( c)  < 0 \  ( c- \epsilon < x < c);
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090790/s09079024.png" /></td> </tr></table>
+
$$
 +
f _ {0} ( x) f _ {1} ( c)  > 0 \  ( c < x < c + \epsilon ).
 +
$$
  
This theorem was proved by J.Ch. Sturm [[#References|[1]]], who also proposed the following method of constructing a Sturm series for a polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090790/s09079025.png" /> with real coefficients and without multiple roots: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090790/s09079026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090790/s09079027.png" />, and, if the polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090790/s09079028.png" /> are already constructed, then as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090790/s09079029.png" /> one should take minus the remainder occurring in the process of dividing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090790/s09079030.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090790/s09079031.png" />. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090790/s09079032.png" /> will be a non-zero constant.
+
This theorem was proved by J.Ch. Sturm [[#References|[1]]], who also proposed the following method of constructing a Sturm series for a polynomial $  f ( x) $
 +
with real coefficients and without multiple roots: $  f _ {0} ( x) = f ( x) $,  
 +
$  f _ {1} ( x) = f ^ { \prime } ( x) $,  
 +
and, if the polynomials $  f _ {0} ( x) \dots f _ {k} ( x) $
 +
are already constructed, then as $  f _ {k+1} ( x) $
 +
one should take minus the remainder occurring in the process of dividing $  f _ {k-1} ( x) $
 +
by $  f _ {k} ( x) $.  
 +
Here, $  f _ {s} ( x) $
 +
will be a non-zero constant.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.Ch. Sturm,  ''Bull. de Férussac'' , '''11'''  (1829)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.G. Kurosh,  "Higher algebra" , MIR  (1972)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.Ch. Sturm,  ''Bull. de Férussac'' , '''11'''  (1829)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.G. Kurosh,  "Higher algebra" , MIR  (1972)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090790/s09079033.png" /> can be described as follows:
+
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 $  f _ {0} ( x) $
 +
can be described as follows:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090790/s09079034.png" /></td> </tr></table>
+
$$
 +
f _ {0} ( x)  = f ( x),\  f _ {1} ( x)  = f ^ { \prime } ( x) ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090790/s09079035.png" /></td> </tr></table>
+
$$
 +
f _ {0} ( x)  = q _ {1} ( x) f _ {1} ( x) - f _ {2} ( x)
 +
,\  \mathop{\rm deg}  f _ {2} ( x) < \mathop{\rm deg}  f _ {1} ( x) ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090790/s09079036.png" /></td> </tr></table>
+
$$
 +
\dots \dots \dots \dots
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090790/s09079037.png" /></td> </tr></table>
+
$$
 +
f _ {k-} 1 ( x)  = q _ {k} ( x) f _ {k} ( x) - f _ {k+1} ( x)
 +
,\  \mathop{\rm deg}  f _ {k+1} ( x) < \mathop{\rm deg}  f _ {k} ( x),
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090790/s09079038.png" /></td> </tr></table>
+
$$
 +
\dots \dots \dots \dots
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090790/s09079039.png" /></td> </tr></table>
+
$$
 +
f _ {s-1} ( x)  = q _ {s} ( x) f _ {s} ( x) ,
 +
$$
  
so <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090790/s09079040.png" /> is a non-zero constant.
+
so $  f _ {s} ( x) $
 +
is a non-zero constant.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Jacobson,  "Basic algebra" , '''I''' , Freeman  (1974)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L.E.J. Dickson,  "New first course in the theory of equations" , Wiley  (1939)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  B.L. van der Waerden,  "Algebra" , '''1''' , Springer  (1967)  (Translated from German)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Jacobson,  "Basic algebra" , '''I''' , Freeman  (1974)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L.E.J. Dickson,  "New first course in the theory of equations" , Wiley  (1939)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  B.L. van der Waerden,  "Algebra" , '''1''' , Springer  (1967)  (Translated from German)</TD></TR></table>

Latest revision as of 10:42, 19 February 2021


If

$$ \tag{* } f _ {0} ( x), \ldots, f _ {s} ( x) $$

is a Sturm series on the interval $ [ a, b] $, $ a < b $, and $ w( x) $ is the number of variations of sign in the series (*) at a point $ x \in [ a, b] $ (vanishing terms are not taken into consideration), then the number of distinct roots of the function $ f _ {0} $ on the interval $ [ a, b] $ is equal to the difference $ w( a)- w( b) $.

A Sturm series (or Sturm sequence) is a sequence of real-valued continuous functions (*) on $ [ a, b] $ having a finite number of roots on this interval, and such that

1) $ f _ {0} ( a) f _ {0} ( b) \neq 0 $;

2) $ f _ {s} ( x) \neq 0 $ on $ [ a, b] $;

3) from $ f _ {k} ( c) = 0 $ for some $ k $ $ ( 0 < k < s) $ and given $ c $ in $ [ a, b] $ it follows that $ f _ {k-1} ( c) f _ {k+1} ( c) < 0 $;

4) from $ f _ {0} ( c) = 0 $ for a given $ c $ $ ( a < c < b) $ it follows that for sufficiently small $ \epsilon > 0 $,

$$ f _ {0} ( x) f _ {1} ( c) < 0 \ ( c- \epsilon < x < c); $$

$$ f _ {0} ( x) f _ {1} ( c) > 0 \ ( c < x < c + \epsilon ). $$

This theorem was proved by J.Ch. Sturm [1], who also proposed the following method of constructing a Sturm series for a polynomial $ f ( x) $ with real coefficients and without multiple roots: $ f _ {0} ( x) = f ( x) $, $ f _ {1} ( x) = f ^ { \prime } ( x) $, and, if the polynomials $ f _ {0} ( x) \dots f _ {k} ( x) $ are already constructed, then as $ f _ {k+1} ( x) $ one should take minus the remainder occurring in the process of dividing $ f _ {k-1} ( x) $ by $ f _ {k} ( x) $. Here, $ f _ {s} ( x) $ 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 $ f _ {0} ( x) $ can be described as follows:

$$ f _ {0} ( x) = f ( x),\ f _ {1} ( x) = f ^ { \prime } ( x) , $$

$$ f _ {0} ( x) = q _ {1} ( x) f _ {1} ( x) - f _ {2} ( x) ,\ \mathop{\rm deg} f _ {2} ( x) < \mathop{\rm deg} f _ {1} ( x) , $$

$$ \dots \dots \dots \dots $$

$$ f _ {k-} 1 ( x) = q _ {k} ( x) f _ {k} ( x) - f _ {k+1} ( x) ,\ \mathop{\rm deg} f _ {k+1} ( x) < \mathop{\rm deg} f _ {k} ( x), $$

$$ \dots \dots \dots \dots $$

$$ f _ {s-1} ( x) = q _ {s} ( x) f _ {s} ( x) , $$

so $ f _ {s} ( x) $ 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)
How to Cite This Entry:
Sturm theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sturm_theorem&oldid=17606
This article was adapted from an original article by I.V. Proskuryakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article