On existence of Budaghyan-Carlet APN hexanomials

More generally, if GCD(m,n)=k⩾1, Budaghyan and Carlet showed that the nonzero derivatives of the hexanomials are 2k-to-one maps from Fr2 to Fr2, provided the same technical condition holds. We prove that their construction produces polynomials with this property for all m and n.