Finiteness of the Hilbert number

Determine whether the Hilbert number H(d), defined as the supremum of the number of isolated periodic orbits of real planar polynomial vector fields of degree at most d, is finite for every polynomial degree d.

Background

The paper studies the second part of Hilbert's sixteenth problem, which concerns the number of limit cycles that planar polynomial vector fields can possess. The Hilbert number H(d) is defined as the supremum of the number of isolated periodic orbits among all real planar polynomial vector fields whose component polynomials have degree at most d.

Although the paper proves new asymptotic lower bounds for H(d), it does not resolve the fundamental question of whether this supremum is finite for each fixed degree. This issue concerns the existence of a general upper-bound theory and is distinct from the paper's constructive lower-bound results.