Generic complex polynomial vector fields with real coefficients

The paper’s Theorem 7.4 states precisely the realization and parametrization of generic real polynomial vector fields of degree k+1 by a modulus (τ, η), with τ a non-crossing involution satisfying the real-symmetry constraints and η living in H^{(k−m+1)/2 − ℓ} × (iR+)^{m−1+ℓ} × (R+)^{ℓ}, together with uniqueness of the monic centered realizing polynomial and the counts of m real singular points and ℓ homoclinic loop(s) along R. The proof proceeds by gluing horizontal strips in rectifying time, uniformizing the resulting surface, and imposing an antiholomorphic involution to ensure real coefficients; this matches the candidate’s Douady–Sentenac/Branner–Dias strip-model approach and reality constraints. Key ingredients in the paper include the analytic invariant for complex-generic vector fields via horizontal strip heights and uniqueness (DES/BD), the real-form combinatorics, the count of real-symmetric strips and pairs, and the existence/uniqueness realization in the real case via a conjugation on the glued surface leading to a monic polynomial of degree k+1 with real coefficients. The only minor difference is expository: the paper describes the glued surface as a sphere with k+1 punctures before push-forward, whereas the candidate speaks of a simply connected parabolic surface; these are compatible viewpoints of the same uniformization step and do not affect correctness. See Theorem 7.4 and its proof outline, the combinatorial/analytic invariant setup, and the gluing and conjugation construction in the paper .