Applications of algebraic methods in solving the center-focus problem

M. N. Popa ; V. V. Pricop

Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 1 (2013), pp. 45-71.

Résumé

The nonlinear differential system $\dot{x}=\sum _{i=0}^{\ell }{P}_{m_i}(x,y)$, $\dot{y}=\sum _{i=0}^{\ell }{Q}_{m_i}(x,y)$ is considered, where $P_{m_i}$ and $Q_{m_i}$ are homogeneous polynomials of degree $m_i\ge 1$ in $x$ and $y$, $m_0=1$. The set $\{1,m_i{\}}_{i=1}^{\ell }$ consists of a finite number $(l\mathrm{\infty })$ of distinct integer numbers. It is shown that the maximal number of algebraically independent focal quantities that take part in solving the center-focus problem for the given differential system with $m_0=1$, having at the origin of coordinates a singular point of the second type (center or focus), does not exceed $\varrho =2(\sum _{i=1}^{\ell }{m}_i+\ell )+3$. We make an assumption that the number $\omega$ of essential conditions for center which solve the center-focus problem for this differential system does not exceed $\varrho$, i.e. $\omega \le \varrho$.

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