Minimal polynomial density with respect to its Newton polytope

Establish that for every polynomial dynamical system of the form x' = g(μ, x) and polynomial observation y = f(μ, x), where g ∈ K[μ, x]^n and f ∈ K[μ, x], the minimal elimination polynomial f_min in the prime ideal I_{g,f} ∩ K[μ, y^{(∞)}] is dense with respect to its Newton polytope in the variables {μ_1,…,μ_r, y, y',…, y^{(ν)}}, meaning that every integer lattice point of the Newton polytope NP(f_min) corresponds to a monomial with a nonzero coefficient in f_min (here ν = ord f_min).

Background

The paper studies differential elimination for polynomial dynamical systems in state-space form with a scalar polynomial observation. The central object is the minimal polynomial f_min of the elimination ideal I_{g,f} ∩ K[μ, y^{(∞)}], which uniquely characterizes the ideal and governs the differential relation satisfied by the measured output y.

Using Theorem 1, the authors bound the support of f_min via a Newton polytope that incorporates degrees in the state variables and parameters, and they experimentally compare the predicted support with the actual support of f_min. Across all reported experiments (both non-parametric and parametric), they observe that f_min is dense with respect to its own Newton polytope, i.e., all lattice points inside NP(f_min) appear as monomials of f_min.

Motivated by consistent empirical evidence, they formulate a conjecture asserting that this density phenomenon holds generally, which, if proved, would validate Newton polytope methods as an accurate tool for support prediction in differential elimination for this class of systems.