Tightness of the main degree bound for D‑algebraic elimination

Determine how tight the upper bound k > (r+1)(d^{1+(r_min−r_l)/(r−r_min+1)} − 1) is for the minimal degree of a nonzero polynomial of order r in the elimination ideal ⟨P_1,…,P_n⟩^{(r−r_l)} ∩ K[y_l,y_l',…], where P_1,…,P_n ∈ R_{r_1,…,r_n} are chosen with r_i = max_j ord_i(P_j), at least one P_j depends on each y_i, d = ∏_{j=1}^n deg(P_j), and the tuple (P_1,…,P_n) is D‑regular at order r−r_l. Ascertain whether this bound is sharp or can be reduced in typical or worst‑case instances.

Background

Theorem \ref{main_theorem} provides a general upper bound on the degree of a nonzero relation obtained by eliminating all variables except one from a D‑regular system of differential polynomials. Because the bound is exponential in parameters and experiments for the exact setting are computationally infeasible, the authors could not empirically assess its sharpness.

Understanding the tightness of this bound is important for predicting the practical size of closure property outputs for D‑algebraic functions and for guiding algorithmic design in nonlinear elimination.