Hilbert's sixteenth problem

As of 2026 Hilbert's sixteenth problem remains open. In its dynamical second part (also Smale's 13th problem), the individual finiteness question — whether a single polynomial vector field $\dot{x}=P(x,y)$, $\dot{y}=Q(x,y)$ can have infinitely many limit cycles — was settled in the negative by Yulij Ilyashenko (1991) and Jean Écalle (1992), who overcame the 1981 gap in Dulac's 1923 memoir by proving that limit cycles cannot accumulate on a polycycle. However, the uniform bound $H(n)$ — the supremum of the number of limit cycles over all polynomial systems of degree $n$ — is not known to be finite for any $n \ge 2$. Even for quadratic systems ($n=2$), where Shi Songling (1980) constructed an example with 4 limit cycles and $H(2)=4$ is widely conjectured, the Dumortier–Roussarie–Rousseau program reducing $H(2)<\infty$ to proving finite cyclicity of 121 degenerate graphics in the Poincaré sphere has not been completed. Lower bounds for general $n$ grow at least like $H(n) = \Omega(n^2 \log n)$ (Christopher–Lloyd, 1995). In the real-algebraic first part, isotopic configurations of nonsingular plane curves are fully classified up to degree $n \le 7$, while degree $8$ and higher remain open.