Differential uniformity and second order derivatives for generic polynomials

Abstract: For any polynomial $f$ of ${\mathbb F}_{2^n}[x]$ we introduce the following characteristic of the distribution of its second order derivative,which extends the differential uniformity notion:$$\delta^2(f):=\max_{\substack{\alpha \in {\mathbb F}_{2^n}^{\ast} ,\alpha' \in {\mathbb F}_{2^n}^{\ast} ,\beta \in {\mathbb F}_{2^n} \alpha\not=\alpha'}} \sharp{x\in{\mathbb F}_{2^n} \mid D_{\alpha,\alpha'}^2f(x)=\beta}$$where $D_{\alpha,\alpha'}^2f(x):=D_{\alpha'}(D_{\alpha}f(x))=f(x)+f(x+\alpha)+f(x+\alpha')+f(x+\alpha+\alpha')$ 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.