||In 1900, Hilbert presented a list consisting of 23 mathematical problems (see ). The second part of the 16th problem appears to be one of the most persistent in that list, second only to the 8th problem, the Riemann conjecture. The second part of the 16th problem is traditionally split into three parts (see ).
Problem 1. A limit cycle is an isolated closed orbit. Is it true that a planar polynomial vector field has but a finite number of limit cycles?
Problem 2. Is it true that the number of limit cycles of a planar polynomial vector field is bounded by a constant depending on the degree of the polynomials only?
Denote the degree of the planar polynomial vector field by n. The bound on the number of limit cycles in Problem 2 is denoted by H(n), and is known as the Hilbert number. Linear vector fields have no limit cycles, hence H( 1) = 0.