Hilbert's Sixteenth Problem (Part II)

Let H(n) be the maximum number of limit cycles of a planar vector field whose components are polynomials of degree at most n. Determine H(2), and in general whether H(n) is finite for each n.

The second half of Hilbert's 16th asks exactly this. Dulac's 1923 claimed finiteness proof stood until Écalle and Ilyashenko independently identified and repaired its gap around 1991, yielding local finiteness per polygonal annulus. The global function H(n): even H(2) is known only within 3 ≤ H(2) ≤ 4, and the countryside between perturbation methods and algebraic geometry remains contested.