We consider a general nonlinear dispersive equation with monomial nonlinearity of order k over \(\mathbb {R}^d\) . We construct a rigorous theory which states that higher-order nonlinearities and higher dimensions induce sharper local well-posedness theories. More precisely, assuming that a certain positive multiplier estimate holds at order \(k_0\) and in dimension \(d_0\) , we prove a sharp local well-posedness result in \(H^s(\mathbb {R}^d)\) for any \(k\geqslant k_0\) and \(d\geqslant d_0\) . Moreover, we give an explicit bound on the gain of regularity observed in the difference between the linear and nonlinear solutions, confirming the conjecture made in [7]. The result is then applied to generalized Korteweg-de Vries, Zakharov-Kuznetsov and nonlinear Schrödinger equations.
