Lower bounds for the Hilbert number of polynomial systems (Q665987)

Consider the planar polynomial systems \[ {dx\over dt}= P_n(x,y),\qquad{dy\over dt}= Q_n(x,y)\tag{E\(_n\)} \] and \[ {dx\over dt}= H_y(x,y)+\varepsilon P_n(x,y),\qquad {dy\over dt}=- H_x(x,y)+ \varepsilon Q_n(x,y),\tag{E\(^*_n\)} \] where \(P_n\) and \(Q_n\) are polynomials in \(x\) and \(y\) of degree \(n\), \(H\) is a polynomial in \(x\) and \(y\) of degree \(m\) with \(m\leq n+1\), \(\varepsilon\) is a small parameter. Let \(H(n)\) and \(H^*(n)\) denote the maximal number of limit cycles of system \((\text{E}_n)\) and \((\text{E}^*_n)\), respectively. The authors give a survey on improved lower bounds for \(H(n)\) and \(H^*(n)\). In particular, they prove \[ \begin{aligned} H(m)\geq m^2\quad &\text{for }m\geq 3,\;m\neq 6,\;m\neq 14,\\ H^*(m)\geq m^2\quad &\text{for }m\geq 23,\\ H^*(2k+1)\geq (2k+1)^2\quad &\text{for }k\geq 1,\end{aligned} \] \(H(m)\) grows at least as \({1\over 2\ln 2}(m+ 2)^2\ln(m+ 2)\).