Applications of algebraic methods in solving the center-focus problem (Q2873824)

The nonlinear differential system NEWLINE\[NEWLINE\dot x=\sum^l_{i=0} P_{m_i}(x,y),\quad \dot y= \sum^l_{i=0} Q_{m_i}(x,y)NEWLINE\]NEWLINE with imaginary roots of the characteristic equation at the singular point \(O(0;0)\) is considered, where \(P_{m_i}\) and \(Q_{m_i}\) are homogeneous polynomials of degree \(m_i\geq 1\) in \(x\) and \(y\), \(m_0=1\). The set \(\{1,m_i\}^l_{i=1}\) consists of a finite number \((l<\infty)\) of distinct integer numbers. It is proved 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 NEWLINE\[NEWLINE\rho=2\Biggl(\sum^l_{i=1} m_i+ l\Biggr)+3.NEWLINE\]