Explain the discrepancy between theoretical and experimental bounds for algebraic relations among polynomials

Investigate and explain the discrepancy between the theoretical guarantee that n+1 polynomials P_0,…,P_n of degree d in K[x_1,…,x_n] admit an algebraic relation of total degree k ≥ (n+1)(d^n − 1) (via the inequality \binom{n+1+k}{n+1} > \binom{n+kd}{n}) and experimental observations suggesting such a relation already exists for k ≥ d^n. Determine the mechanism or refined bounds that reconcile these results.

Background

To gain intuition about the tightness of their main bound, the authors paper a related classical elimination problem: algebraic dependence among n+1 polynomials of fixed degree in n variables. A counting argument yields a sufficient condition on the relation degree, but experiments indicate a significantly smaller typical threshold.

Clarifying this gap could inform improvements to degree bounds in the D‑algebraic setting and indicate whether current counting-based estimates systematically overshoot.