Constructing rational Lyapunov certificates for planar homogeneous fields

For each fixed odd degree d, we give a deterministic construction of strict rational Lyapunov functions for globally asymptotically stable planar homogeneous polynomial vector fields with rational coefficients. If B is the dense binary input length, construction time, degrees and total dense output length are bounded by 2^{C_d(B+1)} for an effectively computable constant C_d. The output is P/(x^2+y^2)^r, with P positive homogeneous of degree 2r+2, and extends C1 to the origin. The construction covers nonvanishing angular velocity, angular sign changes and zeros of arbitrary multiplicity. It combines uniform algebraic separation for the rotational mean, normalized nonnegative trigonometric powers and a finite-power conversion. Rational converse theorems and planar stability decision are prior results. No general software implementation, sharp bound, polynomial-certificate theorem or higher-dimensional conclusion is asserted.