Dembowski–Ostrom polynomials from Dickson polynomials

Motivated by several recent results, we determine precisely when Fk(Xd,a)−Fk(0,a) is a Dembowski–Ostrom polynomial, where Fk(X,a) is a Dickson polynomial of the first or second kind. As a consequence, we obtain a classification of all such polynomials which are also planar; all examples found are equivalent to previously known examples.