Asymptotic tightness of the support bound as degrees increase

Establish that, for polynomial dynamical systems x' = g(μ, x), y = f(μ, x) with fixed dimension and parameter degrees, the support bound given by Theorem 1 becomes tight in the limits deg_x f → ∞ or max_i deg_x g_i → ∞, i.e., the number of monomials in the minimal polynomial f_min equals the number of integer lattice points predicted by the bound, equivalently NP(f_min) coincides with the bound’s Newton polytope in these asymptotic regimes.

Background

The main theoretical result (Theorem 1) provides a multi-homogeneous Newton polytope bound on the support of the minimal polynomial f_min for elimination in polynomial ODE models with parameters. The experimental section quantifies the bound’s accuracy as the ratio of the number of monomials in f_min to the number of monomials predicted by the bound.

Empirical data shows that this accuracy improves toward 100% as either the degree of the observation polynomial in the state variables (d_x) or the maximum degree of the dynamics in the state variables (D_x) increases. Based on this empirical trend, the authors conjecture asymptotic tightness of their support bound in these limits.

Proving this conjecture would demonstrate that the bound not only accurately predicts support generically but becomes exact for large degrees, strengthening the theoretical foundation of the evaluation–interpolation elimination approach.