Applications of algebraic methods in solving the center-focus problem

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 $m_i\geq1$ in $x$ and $y$, $m_0=1$. The set $\{1,m_i\}_{i=1}^\ell$ consists of a finite number $(l<\infty)$ 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. $\omega\leq\varrho$.