Kronecker formula

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