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The second part of Hilbert's 16th problem deals with polynomial dierential equations in the plane. It remains unsolved even for quadratic polynomials. There were several attempts to solve it that failed. Yet the prob- lem inspired signicant progress in the geometric theory of planar dierential 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, are presented below. x1. The problem and its counterparts What may be said about the number and location of limit cycles of a planar polynomial vector eld of degree n? (The limit cycle is an isolated closed orbit of a vector eld.) This second part of Hilbert's 16th problem appears to be one of the most persistent in the famous Hilbert list (H), second only to the Riemann -function conjecture. Traditionally, Hilbert's question is split into three, each one requiring a stronger answer. Problem 1. Is it true that a planar polynomial vector eld has but a nite number of limit cycles?

Centennial History of Hilbert's 16th Problem