The Existential Hilbert 16-th problem and an estimate for cyclicity of elementary polycycles
The Existential Hilbert Problem is a weak version of the part b of the Hilbert 16-th problem which also asks not only about the number, but also about position of limit cycles of (1). The problem about finiteness of number of limit cycles for an individual polynomial line field (1) is called Dulac problem, since the pioneering work of Dulac [Du], who claimed in 1923 to solve this problem, but an error was found. The Dulac problem was solved by two independent and rather different proofs given independently