Sharp bounds for minimal differential relations when d > D

Establish sharp bounds, expressed solely in terms of n, d = deg(g1), and D = max_{2≤i≤n} deg(g_i), for the Newton polytope of the minimal differential relation δ ∈ C[x1, x1', ..., x1^{(n)}] satisfied by x1 in polynomial dynamical systems of the form x'_i = g_i(x1, ..., xn), in the regime where d > D. Specifically, determine tight bounds that are best possible (sharp) for the size and shape of the Newton polytope of δ when deg(g1) exceeds the maximum degree among g2, ..., gn.

Background

The paper considers differential elimination for polynomial dynamical systems x'i = g_i(x1, ..., xn). For such systems, the coordinate x1 of a solution vector (x1, ..., xn) satisfies a minimal differential relation δ ∈ C[x1, x1', ..., x1^{(n)}]. Prior work constructs a polytope A_min that contains the Newton polytope of δ by giving explicit inequalities for the support; these bounds depend on the number of variables n and on polynomial degree parameters d = deg(g1) and D = max{2≤i≤n} deg(g_i).

The authors note that the bounds were shown to be sharp when d ≤ D, but that achieving sharp bounds for the complementary regime d > D remains unresolved. They investigate whether mixed fiber polytopes can provide practical improvements, yet the theoretical question of sharp bounds in the d > D case is explicitly left open.