A GLOBAL ISOCHRONOUS CENTER IS LINEAR

Villarini’s Theorem 1.1 rigorously proves that a planar polynomial vector field with a center at infinity whose period function is bounded below must be linear, and after an affine change of variables and time-rescaling it becomes the standard rotation field −y ∂/∂x + x ∂/∂y; see the theorem statement and conclusion in the paper and the closing argument of the proof. The candidate solution reaches the same conclusion via a different route: it works in polar coordinates, analyzes the highest-degree homogeneous part to deduce θ̇ ≳ r^{n−1}, and shows that, unless n = 1, periods of outer closed orbits would tend to zero, contradicting the assumed lower bound. This is a sound approach in essence. Two caveats: (i) the candidate’s claim that any neighborhood of an equilibrium contains nonperiodic trajectories is stated without qualification (a center is a counterexample in general), though in the specific compactified-at-infinity setting the equilibria on the boundary circle s = 0 are never centers, so a brief linearization argument would close this gap; and (ii) the dicritical case R_n ≡ 0 is not addressed explicitly by the candidate, while the paper treats it and excludes it under the hypotheses.