For every degree, we prove that the number of isolated periodic orbits of a real planar polynomial vector field is bounded by a finite constant depending only on that degree. This establishes the uniform boundedness assertion in the second part of Hilbert's sixteenth problem. The proof uses separation of asymptotic expansions on nested complex domains and a finite-dimensional counting argument.

Every classical Liénard system $\dot x=y-F(x)$, $\dot y=-x$, with F an arbitrary real polynomial of degree at most five, has at most two geometrically distinct isolated periodic orbits, and the bound is attained. This proves the degree-five case of the Lins Neto–de Melo–Pugh conjecture.