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 , , and be real-valued continuously differentiable functions on such that the system of equations

(1)

has a finite number of roots. Suppose that the equation

defines a closed surface not passing through the roots of the system (1), and that in the interior of . If the functions , , are considered as components of a vector field on , then their singular points (by definition) coincide with the roots of the system (1). Let be some root and let be its index as a singular point (cf. Singular point, index of a). Then

(2)

(summation over all roots), where is the surface area of the unit sphere ,

and, if is any function, denotes the derivative . Formula (2) is Kronecker's formula.

If , the space integral in (2) disappears, and one obtains an expression for the sum of the indices of the singular points of the vector field in the interior of the surface , i.e. an expression for the degree of the mapping from the surface into the sphere obtained by restricting the mapping , , to . Under certain additional assumptions, is equal to the so-called Kronecker characteristic of the system of functions (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