Differential uniformity and second order derivatives for generic polynomials

For any polynomial f of F2n[x] we introduce the following characteristic of the distribution of its second order derivative, which extends the differential uniformity notion:δ2(f):=maxα∈F2n⁎,α′∈F2n⁎,β∈F2nα≠α′⁡♯{x∈F2n|Dα,α′2f(x)=β} where Dα,α′2f(x):=Dα′(Dαf(x))=f(x)+f(x+α)+f(x+α′)+f(x+α+α′) 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.