Limit-Cycle Replication via Chebyshev Pullbacks and a Quadratic Ceiling for Separable Schemes

Abstract

Let \(H(n)\) denote the Hilbert number, i.e.\ the maximal number of limit cycles of planar polynomial vector fields of degree \(\le n\). A classical lower-bound mechanism for \(H(n)\) is \emph{replication}: one pulls back a vector field by a polynomial map and lifts each existing limit cycle to several disjoint copies while controlling the resulting degree. In this paper we give a fully self-contained replication theorem based on the separable Chebyshev covering \[ \Phi(u,v)=(T_m(u),T_m(v)). \] Using the \(m\) monotone full branches of \(T_m\) on \((-1,1)\), we prove that every degree-\(\le n\) polynomial vector field with \(k\) limit cycles gives rise to a degree-\(\le nm+m-1\) polynomial vector field with at least \(m^2k\) limit cycles. Consequently, \[ H(nm+m-1)\ge m^2H(n)\qquad (m\ge 2). \] We then extend the construction to general separable pullbacks \((u,v)\mapsto (p(u),p(v))\), show that Chebyshev attains the maximal possible branch count among degree-\(m\) separable pullbacks, and prove a quadratic ceiling for replication-only schemes: if one iterates separable pullbacks and no additional limit cycles are created beyond those forced by lifting, then the number of resulting limit cycles is at most quadratic in the final degree. This shows that superquadratic lower bounds, such as the known \(n^2\log n\)-type bounds, necessarily require mechanisms beyond pure separable replication. Finally, combining our replication theorem with the strongest currently published seed bounds, we obtain new explicit lower estimates in several degrees, including \begin{gather*} H(14)\ge 252,\qquad H(29)\ge 1080,\\ H(31)\ge 1380,\qquad H(39)\ge 2012. \end{gather*}