GENERIC REVERSIBLE COMPLEX POLYNOMIAL VECTOR FIELDS

The paper proves three main statements for reversible polynomial vector fields v(z)=iP(z)∂/∂z with P real, monic, centered: (i) a classification of generic strata by non‑crossing involutions on {0,…,k} that preserve intervals between fixed points (Theorem 3.3), with the number of fixed points equal to the number of real centers and with q minimal τ‑invariant blocks satisfying q=h+1 where h is the number of symmetric homoclinic loops (Proposition 3.5); (ii) the analytic modulus of a generic stratum is η∈(R+)^h×(iR+)^(h+1−m)×H^{(k−2h+m−1)/2} (Theorem 3.11); and (iii) every admissible pair (τ,η) is realized by a unique vector field in PRev,k+1 (Theorem 7.1). All three are explicitly stated and proved in the uploaded paper. The candidate solution matches (i) and (iii) at a high level but misidentifies the analytic invariants in (ii): it incorrectly assigns the H-factors to center (rotation) zones and the R+-factors to αω-strips “between blocks.” In the paper, R+ parameters come from rotation zones (lengths κ>0 in the rectified coordinate), iR+ from αω-zones that intersect the real axis (transversal times just above R are purely imaginary), and H from αω-zones contained in the upper half-plane; see Theorem 3.11 and the discussion/figures around it. This misattribution also leads to a wrong statement that the number of center zones equals (k−2h+m−1)/2, whereas that exponent counts αω-zones contained in the upper half-plane. The uniqueness/existence and the combinatorial classification are otherwise consistent with the paper’s statements. Citations: classification via a non-crossing involution preserving intervals between fixed points (Theorem 3.3) ; relation m=#Fix(τ), q=h+1 (Proposition 3.5) ; analytic modulus η and its factorization (Theorem 3.11 and Definition 3.13) ; realization and uniqueness (Theorem 7.1 and its proof) ; construction details with antiholomorphic symmetry in the gluing argument .