ON THE NUMBER OF ISOLATED INVARIANT TORI FOR 3D POLYNOMIAL VECTOR FIELDS

The paper’s Theorem A states N(m) ≥ Hh([m/2] − 1) for all m ≥ 2, proved by constructing, from any planar degree-k system with H hyperbolic limit cycles in K, a one-parameter family of 3D polynomial systems (ẋ, ẏ, ż) = (−y, x + ε y P(x^2 + y^2, z), 2ε y^2 Q(x^2 + y^2, z)) having at least H normally hyperbolic invariant tori; when deg P, deg Q ≤ k, this 3D system has degree 2k+2, yielding N(2k+2) ≥ Hh(k) and, by monotonicity, the odd-m case as well . The averaging step and guiding system equivalence are carried out explicitly via cylindrical coordinates, time rescaling, and the transformation ρ = r^2, reducing to the original planar system’s limit cycles; Theorem 1 then yields the invariant tori near those cycles . The candidate solution reproduces this mechanism verbatim: fix n = [m/2] − 1, lift planar degree ≤ n systems to 3D systems of degree ≤ 2(n+1) ≤ m, and take suprema over planar systems. Minor overstatements aside (e.g., saying “one-to-one correspondence” rather than “at least H”), the logic, degree control, and use of averaging are aligned with the paper’s proof.