Abstract:
Two classical problems on plane polynomial vector fields, Hilbert's 16th problem about the maximal number of limit cycles in such a system and Poincaré's center-focus problem about conditions for all trajectories around a critical point to be closed, can be naturally reformulated for the Abel differential equation $y'=p(x)y^2+q(x)y^3$. Recently, the center conditions for the Abel equation have been related to the composition factorization of $P=\int p$ and $Q=\int q$ and to the vanishing conditions for the moments $m_{i,j}=\int P^i Q^j q$.
On the basis of these results we start in the present paper the investigation of the “Hilbert's tangential problem” for the Abel equation, which is to find a bound for the number of zeroes of $I(t) =\int^b_a(q(x)dx)/(1-tP(x))$.
Key words and phrases:Limit cycles, Abel differential equation, moments, compositions, Bautin ideals.
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