# Kronecker formula

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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