On the 16th Hilbert problem for limit cycles on nonsingular algebraic curves (Q619876)

Consider the planar polynomial system \[ {dx\over dt}= P(x,y),\quad {dy\over dt}= Q(x,y)\tag{\(*\)} \] of degree \(n\). Suppose that \((*)\) exactly \(k\) invariant algebraic curves \(g_m(x,y)= 0\), \(1\leq m\leq k\). Denote by \(A(k,n)\) the maximum number of algebraic limit cycles of \((*)\). The authors prove the following results. If the invariant algebraic curves \(g_m(x,y)= 0\), \(m= 1,\dotsc, k\), of \((*)\) are nonsingular and irreducible, then \[ A(k,n)\leq k\Biggl({n(n-1)\over 2}+ 1\Biggr)\leq {n^4\over 4}+ {3n^2\over 4}+ 1. \] If the curves \(g_m(x,y)= 0\), \(m= 1,\dotsc, k\), are irreducible and \(\sum^k_{m=1} \deg g_m\leq n+1\), then \[ A(k,n)\leq \begin{cases} {n(n-1)\over 2}+ 1\quad &\text{if }n\text{ is even},\\ {n(n-1)\over 2}\quad &\text{if }n\text{ is odd}.\end{cases} \] Additionally, sufficient conditions are provided that all algebraic limit cycles are hyperbolic.