Namespaces
Variants
Actions

Difference between revisions of "Kronecker formula"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
A formula for the algebraic sum of the values of a function on the set of roots of a system of equations; established by L. Kronecker , [[#References|[2]]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055870/k0558701.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055870/k0558702.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055870/k0558703.png" /> be real-valued continuously differentiable functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055870/k0558704.png" /> such that the system of equations
+
<!--
 +
k0558701.png
 +
$#A+1 = 34 n = 0
 +
$#C+1 = 34 : ~/encyclopedia/old_files/data/K055/K.0505870 Kronecker formula
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
<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/k/k055/k055870/k0558705.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
 +
A formula for the algebraic sum of the values of a function on the set of roots of a system of equations; established by L. Kronecker , [[#References|[2]]]. Let  $  F ^ { t } ( x  ^ {1} \dots x  ^ {n} ) $,
 +
$  t = 0 \dots n $,
 +
and  $  f ( x  ^ {1} \dots x  ^ {n} ) $
 +
be real-valued continuously differentiable functions on  $  \mathbf R  ^ {n} $
 +
such that the system of equations
 +
 
 +
$$ \tag{1 }
 +
F ^ { s } ( x  ^ {1} \dots x  ^ {n} ) =  0,\ \
 +
s = 1 \dots n,
 +
$$
  
 
has a finite number of roots. Suppose that the equation
 
has a finite number of roots. Suppose that the equation
  
<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/k/k055/k055870/k0558706.png" /></td> </tr></table>
+
$$
 +
F ^ { 0 } ( x  ^ {1} \dots x  ^ {n} )  = 0
 +
$$
 +
 
 +
defines a closed surface  $  P $
 +
not passing through the roots of the system (1), and that  $  F ^ { 0 } < 0 $
 +
in the interior of  $  P $.
 +
If the functions  $  F ^ { s } $,
 +
$  s = 1 \dots n $,
 +
are considered as components of a vector field on  $  \mathbf R  ^ {n} $,
 +
then their singular points (by definition) coincide with the roots of the system (1). Let  $  x _  \alpha  $
 +
be some root and let  $  \chi ( x _  \alpha  ) $
 +
be its index as a singular point (cf. [[Singular point, index of a|Singular point, index of a]]). Then
 +
 
 +
$$ \tag{2 }
  
defines a closed surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055870/k0558707.png" /> not passing through the roots of the system (1), and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055870/k0558708.png" /> in the interior of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055870/k0558709.png" />. If the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055870/k05587010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055870/k05587011.png" />, are considered as components of a vector field on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055870/k05587012.png" />, then their singular points (by definition) coincide with the roots of the system (1). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055870/k05587013.png" /> be some root and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055870/k05587014.png" /> be its index as a singular point (cf. [[Singular point, index of a|Singular point, index of a]]). Then
+
\frac{1}{K  ^ {n} }
  
<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/k/k055/k055870/k05587015.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
\sum _  \alpha
 +
f ( x _  \alpha  )
 +
\chi ( x _  \alpha  ) =
 +
$$
  
<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/k/k055/k055870/k05587016.png" /></td> </tr></table>
+
$$
 +
= \
 +
\int\limits _ {F  ^ {0} < 0 }
 +
\frac \Lambda {R  ^ {n} }
 +
  dV
 +
- \int\limits _ {F  ^ {0} = 0 }
 +
\frac{fD}{QR  ^ {n} }
 +
  dS
 +
$$
  
(summation over all roots), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055870/k05587017.png" /> is the surface area of the unit sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055870/k05587018.png" />,
+
(summation over all roots), where $  K  ^ {n} $
 +
is the surface area of the unit sphere $  S ^ {n - 1 } $,
  
<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/k/k055/k055870/k05587019.png" /></td> </tr></table>
+
$$
 +
= \
 +
\sqrt {\sum _ {s = 1 } ^ { n }  ( F ^ { s } )  ^ {2} } ,\ \
 +
= \
 +
\sqrt {\sum _ {i = 1 } ^ { n }  ( F _ {i} ^ { 0 } )  ^ {2} } ,
 +
$$
  
<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/k/k055/k055870/k05587020.png" /></td> </tr></table>
+
$$
 +
\Lambda  = \left |
 +
\begin{array}{cccc}
 +
0  &f _ {1}  &\dots  &f _ {n}  \\
 +
F ^ { 1 }  &F _ {1} ^ { 1 }  &\dots  &F _ {n} ^ { 1 }  \\
 +
\cdot  &\cdot  &\dots  &\cdot  \\
 +
F ^ { n }  &F _ {n} ^ { 1 }  &\dots  &F _ {n} ^ { n }  \\
 +
\end{array}
 +
\right | ,\  D  = \left |
 +
\begin{array}{cccc}
 +
F ^ { 0 }  &F _ {1} ^ { 0 }  &\dots  &F _ {n} ^ { 0 }  \\
 +
F ^ { 1 }  &F _ {1} ^ { 1 }  &\dots  &F _ {n} ^ { 1 }  \\
 +
\cdot  &\cdot  &\dots  &\cdot  \\
 +
F ^ { n }  &F _ {1} ^ { n }  &\dots  &F _ {n} ^ { n }  \\
 +
\end{array}
 +
\right | ,
 +
$$
  
and, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055870/k05587021.png" /> is any function, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055870/k05587022.png" /> denotes the derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055870/k05587023.png" />. Formula (2) is Kronecker's formula.
+
and, if $  \Phi $
 +
is any function, $  \Phi _ {i} $
 +
denotes the derivative $  \partial  \Phi / \partial  x  ^ {i} $.  
 +
Formula (2) is Kronecker's formula.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055870/k05587024.png" />, the space integral in (2) disappears, and one obtains an expression for the sum of the indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055870/k05587025.png" /> of the singular points of the vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055870/k05587026.png" /> in the interior of the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055870/k05587027.png" />, i.e. an expression for the degree of the mapping from the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055870/k05587028.png" /> into the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055870/k05587029.png" /> obtained by restricting the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055870/k05587030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055870/k05587031.png" />, to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055870/k05587032.png" />. Under certain additional assumptions, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055870/k05587033.png" /> is equal to the so-called Kronecker characteristic of the system of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055870/k05587034.png" /> (see [[#References|[3]]]).
+
If $  f \equiv 1 $,  
 +
the space integral in (2) disappears, and one obtains an expression for the sum of the indices $  \chi _ {F} $
 +
of the singular points of the vector field $  \{ F ^ { s } \} $
 +
in the interior of the surface $  P $,  
 +
i.e. an expression for the degree of the mapping from the surface $  P $
 +
into the sphere $  S ^ {n - 1 } $
 +
obtained by restricting the mapping $  {\widetilde{F}  } {} ^ { s } = F ^ { s } /R $,  
 +
$  s = 1 \dots n $,  
 +
to $  P $.  
 +
Under certain additional assumptions, $  \chi _ {F} $
 +
is equal to the so-called Kronecker characteristic of the system of functions $  F  ^ {0} , F ^ { 1 } \dots F ^ { n } $(
 +
see [[#References|[3]]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1a]</TD> <TD valign="top">  L. Kronecker,  "Ueber Systeme von Funktionen mehrer Variablen. Erster Abhandlung"  ''Monatsberichte''  (1869)  pp. 159–193</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top">  L. Kronecker,  "Ueber Systeme von Funktionen mehrer Variablen. Zweite Abhandlung"  ''Monatsberichte''  (1869)  pp. 688–698</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L. Kronecker,  "Ueber Sturm'sche Funktionen"  ''Monatsberichte''  (1878)  pp. 95–121</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N.G. Chetaev,  "Stability of motion. Studies in analytical mechanics" , Moscow  (1946)  (In Russian)</TD></TR><TR><TD valign="top">[4a]</TD> <TD valign="top">  H. Poincaré,  "Mémoire sur les courbes définiés par une équation differentielle"  ''J. de Math.'' , '''7'''  (1881)  pp. 375–422</TD></TR><TR><TD valign="top">[4b]</TD> <TD valign="top">  H. Poincaré,  "Mémoire sur les courbes définiés par une équation differentielle"  ''J. de Math.'' , '''8'''  (1882)  pp. 251–296</TD></TR><TR><TD valign="top">[4c]</TD> <TD valign="top">  H. Poincaré,  "Mémoire sur les courbes définiés par une équation differentielle"  ''J. de Math.'' , '''1'''  (1885)  pp. 167–244</TD></TR><TR><TD valign="top">[4d]</TD> <TD valign="top">  H. Poincaré,  "Mémoire sur les courbes définiés par une équation differentielle"  ''J. de Math.'' , '''2'''  (1886)  pp. 151–217</TD></TR></table>
 
<table><TR><TD valign="top">[1a]</TD> <TD valign="top">  L. Kronecker,  "Ueber Systeme von Funktionen mehrer Variablen. Erster Abhandlung"  ''Monatsberichte''  (1869)  pp. 159–193</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top">  L. Kronecker,  "Ueber Systeme von Funktionen mehrer Variablen. Zweite Abhandlung"  ''Monatsberichte''  (1869)  pp. 688–698</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L. Kronecker,  "Ueber Sturm'sche Funktionen"  ''Monatsberichte''  (1878)  pp. 95–121</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N.G. Chetaev,  "Stability of motion. Studies in analytical mechanics" , Moscow  (1946)  (In Russian)</TD></TR><TR><TD valign="top">[4a]</TD> <TD valign="top">  H. Poincaré,  "Mémoire sur les courbes définiés par une équation differentielle"  ''J. de Math.'' , '''7'''  (1881)  pp. 375–422</TD></TR><TR><TD valign="top">[4b]</TD> <TD valign="top">  H. Poincaré,  "Mémoire sur les courbes définiés par une équation differentielle"  ''J. de Math.'' , '''8'''  (1882)  pp. 251–296</TD></TR><TR><TD valign="top">[4c]</TD> <TD valign="top">  H. Poincaré,  "Mémoire sur les courbes définiés par une équation differentielle"  ''J. de Math.'' , '''1'''  (1885)  pp. 167–244</TD></TR><TR><TD valign="top">[4d]</TD> <TD valign="top">  H. Poincaré,  "Mémoire sur les courbes définiés par une équation differentielle"  ''J. de Math.'' , '''2'''  (1886)  pp. 151–217</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 22:15, 5 June 2020


A formula for the algebraic sum of the values of a function on the set of roots of a system of equations; established by L. Kronecker , [2]. Let $ F ^ { t } ( x ^ {1} \dots x ^ {n} ) $, $ t = 0 \dots n $, and $ f ( x ^ {1} \dots x ^ {n} ) $ be real-valued continuously differentiable functions on $ \mathbf R ^ {n} $ such that the system of equations

$$ \tag{1 } F ^ { s } ( x ^ {1} \dots x ^ {n} ) = 0,\ \ s = 1 \dots n, $$

has a finite number of roots. Suppose that the equation

$$ F ^ { 0 } ( x ^ {1} \dots x ^ {n} ) = 0 $$

defines a closed surface $ P $ not passing through the roots of the system (1), and that $ F ^ { 0 } < 0 $ in the interior of $ P $. If the functions $ F ^ { s } $, $ s = 1 \dots n $, are considered as components of a vector field on $ \mathbf R ^ {n} $, then their singular points (by definition) coincide with the roots of the system (1). Let $ x _ \alpha $ be some root and let $ \chi ( x _ \alpha ) $ be its index as a singular point (cf. Singular point, index of a). Then

$$ \tag{2 } \frac{1}{K ^ {n} } \sum _ \alpha f ( x _ \alpha ) \chi ( x _ \alpha ) = $$

$$ = \ \int\limits _ {F ^ {0} < 0 } \frac \Lambda {R ^ {n} } dV - \int\limits _ {F ^ {0} = 0 } \frac{fD}{QR ^ {n} } dS $$

(summation over all roots), where $ K ^ {n} $ is the surface area of the unit sphere $ S ^ {n - 1 } $,

$$ R = \ \sqrt {\sum _ {s = 1 } ^ { n } ( F ^ { s } ) ^ {2} } ,\ \ Q = \ \sqrt {\sum _ {i = 1 } ^ { n } ( F _ {i} ^ { 0 } ) ^ {2} } , $$

$$ \Lambda = \left | \begin{array}{cccc} 0 &f _ {1} &\dots &f _ {n} \\ F ^ { 1 } &F _ {1} ^ { 1 } &\dots &F _ {n} ^ { 1 } \\ \cdot &\cdot &\dots &\cdot \\ F ^ { n } &F _ {n} ^ { 1 } &\dots &F _ {n} ^ { n } \\ \end{array} \right | ,\ D = \left | \begin{array}{cccc} F ^ { 0 } &F _ {1} ^ { 0 } &\dots &F _ {n} ^ { 0 } \\ F ^ { 1 } &F _ {1} ^ { 1 } &\dots &F _ {n} ^ { 1 } \\ \cdot &\cdot &\dots &\cdot \\ F ^ { n } &F _ {1} ^ { n } &\dots &F _ {n} ^ { n } \\ \end{array} \right | , $$

and, if $ \Phi $ is any function, $ \Phi _ {i} $ denotes the derivative $ \partial \Phi / \partial x ^ {i} $. Formula (2) is Kronecker's formula.

If $ f \equiv 1 $, the space integral in (2) disappears, and one obtains an expression for the sum of the indices $ \chi _ {F} $ of the singular points of the vector field $ \{ F ^ { s } \} $ in the interior of the surface $ P $, i.e. an expression for the degree of the mapping from the surface $ P $ into the sphere $ S ^ {n - 1 } $ obtained by restricting the mapping $ {\widetilde{F} } {} ^ { s } = F ^ { s } /R $, $ s = 1 \dots n $, to $ P $. Under certain additional assumptions, $ \chi _ {F} $ is equal to the so-called Kronecker characteristic of the system of functions $ F ^ {0} , F ^ { 1 } \dots F ^ { n } $( see [3]).

References

[1a] L. Kronecker, "Ueber Systeme von Funktionen mehrer Variablen. Erster Abhandlung" Monatsberichte (1869) pp. 159–193
[1b] L. Kronecker, "Ueber Systeme von Funktionen mehrer Variablen. Zweite Abhandlung" Monatsberichte (1869) pp. 688–698
[2] L. Kronecker, "Ueber Sturm'sche Funktionen" Monatsberichte (1878) pp. 95–121
[3] N.G. Chetaev, "Stability of motion. Studies in analytical mechanics" , Moscow (1946) (In Russian)
[4a] H. Poincaré, "Mémoire sur les courbes définiés par une équation differentielle" J. de Math. , 7 (1881) pp. 375–422
[4b] H. Poincaré, "Mémoire sur les courbes définiés par une équation differentielle" J. de Math. , 8 (1882) pp. 251–296
[4c] H. Poincaré, "Mémoire sur les courbes définiés par une équation differentielle" J. de Math. , 1 (1885) pp. 167–244
[4d] H. Poincaré, "Mémoire sur les courbes définiés par une équation differentielle" J. de Math. , 2 (1886) pp. 151–217

Comments

Kronecker's characteristic of a system of functions is the origin of the notion of the Degree of a mapping. Cf. [a1] for historical remarks.

References

[a1] M.W. Hirsch, "Differential topology" , Springer (1976) pp. Chapt. 5, Sect. 3
How to Cite This Entry:
Kronecker formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kronecker_formula&oldid=47527
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article