Algebraic Synthesis of Polynomial Vector Fields with Prescribed Invariant Spheres

We introduce a family of polynomial vector fields on the plane generated by factorization in the ring of radial invariants. This family, referred to as the Radial Factorization Family, is generated by coupling an arbitrary monic polynomial \( P \) in the radial invariant \( \Delta=x^{2}+y^{2} \) with the standard rotational generator. It admits an explicit reduction in polar coordinates, yielding a clean global unperturbed model whose periodic orbits are concentric circles with explicitly known radii. By adding a small polynomial perturbation that breaks rotational symmetry, we compute the first-order Melnikov function and establish that the displacement function of the Poincaré return map admits the expansion \( d(r,\varepsilon)=2\pi\varepsilon\,\mathcal{M}(r)+O(\varepsilon^{2}) \) on compact subintervals of the period annulus. Because the unperturbed orbits are circles, the resulting integrals admit explicit evaluation. For the linear center this recovers the classical bound \( \lfloor(d-1)/2\rfloor \) on the number of bifurcating limit cycles; near a hyperbolic invariant circle of a general member of the family the same framework shows unique persistence of a single limit cycle, while in the degenerate case of multiple roots the circles lose normal hyperbolicity and unfold under perturbation into at most \( m \) limit cycles controlled by the multiplicity. The central novelty of the construction lies in its inverse character: in higher dimensions it yields an optimal inverse-realization procedure that converts any prescribed collection of nested invariant spheres—together with a topologically admissible assignment of stability types—into a polynomial vector field of minimal degree within the radial-factorized equivariant class. This provides a systematic algebraic dictionary for topological synthesis, transforming the traditional paradigm of "analyse a given system" into "t design a system with prescribed dynamical skeleton". The framework provides a constructive algebraic perspective related to aspects of Hilbert's sixteenth problem and offers a practical tool for phase-space engineering in nonlinear dynamics.