Differential uniformity and second order derivatives for generic polynomials

For any polynomial $f$ of $\mathbb{F}\mathrm{\_}{2}^n[x]$ we introduce the following characteristic of the distribution of its second order derivative,which extends the differential uniformity notion:$${\delta }^2(f):=\max \mathrm{\_}\begin{array}{c}\alpha \in \mathbb{F}\mathrm{\_}{2^n}^{\ast },{\alpha }^{\mathrm{\prime }}\in \mathbb{F}\mathrm{\_}{2^n}^{\ast },\beta \in \mathbb{F}\mathrm{\_}{2}^n\alpha ≠{\alpha }^{\mathrm{\prime }}\end{array}\mathrm{♯}\{x\in \mathbb{F}\mathrm{\_}{2}^n\mid D\mathrm{\_}{\alpha ,{\alpha }^{\mathrm{\prime }}}^{2}f(x)=\beta \}$$where $D\mathrm{\_}{\alpha ,{\alpha }^{\mathrm{\prime }}}^{2}f(x):=D\mathrm{\_}{\alpha }^{\mathrm{\prime }}(D\mathrm{\_}\alpha f(x))=f(x)+f(x+\alpha )+f(x+{\alpha }^{\mathrm{\prime }})+f(x+\alpha +{\alpha }^{\mathrm{\prime }})$ is the second order derivative.Our purpose is to prove a density theorem relative to this quantity,which is an analogue of a density theorem proved by Voloch for the differential uniformity.