zbMATH — the first resource for mathematics

Calculation of the Kronecker-Poincaré index of algebraic curves with respect to algebraic fields on the plane. (Russian) Zbl 0742.34006
Let \(P\), \(Q\), \(\phi\) be polynomials with real coefficients. Let \(K=\{(x,y)\in\mathbb{R}^ 2,\;\phi(x,y)=0\}\) be a Jordan curve such that on \(K\) there is no singular point of the vector field \((P(x,y),Q(x,y))\). The interior of \(K\) is defined by \(\phi<0\) and \((\partial\phi/\partial x)^ 2+(\partial\phi/\partial y)^ 2\neq 0\) on \(K\). The author gives a purely algebraic algorithm for calculation of the Kronecker-Poincaré index which uses a finite number of elementary algebraic operations of the coefficients of \(P\), \(Q\), \(\phi\).

34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.