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Yulij Ilyashenko

Centennial History of the Hilbert's 16th problem

The second part of the Hilbert 16th problem deals with polynomial differential equations in the plane. It stays unsolved even for quadratic polynomials. There were several attempts to solve it that failed. Yet the problem inspired a significant progress in geometric theory of planar differential equations, as well as bifurcation theory, normal forms, foliations and some topics in algebraic geometry. The dramatic history of the problem as well as related developments is presented below.