Uniform finiteness in Hilbert's sixteenth problem

The second part of Hilbert's sixteenth problem asks whether the number of limit cycles — isolated periodic orbits — of a planar polynomial vector field $\dot x=P(x,y)$, $\dot y=Q(x,y)$ can be bounded uniformly in terms of the degree alone: for each $d\ge2$ there should be a finite $H(d)$ valid for every field with $\max\{\deg P,\deg Q\}\le d$. The problem appeared in Hilbert's celebrated 1900 list [Hilbert1902MathematicalProblems].

The non-uniform half of the question, Dulac's problem, is settled: Écalle and Ilyashenko independently proved that each fixed polynomial field has only finitely many limit cycles [Ilyashenko1991FinitenessLimitCycles]. Those proofs, however, yield no bound depending only on $d$, and no value of $H(d)$ is known even for $d=2$. Quantitative progress has instead concentrated on restricted versions, notably the tangential Hilbert sixteenth problem for perturbations of Hamiltonian fields [Yakovenko2001QuantitativeHilbert].

Recently claimed elementary formulas for $H(d)$ have been examined and explicitly refuted [BuzziNovaes2024HilbertAttempt]. The problem remains open; a resolution requires finiteness arguments that behave uniformly across the whole degree-$d$ family rather than working field by field.