Plane polar Cremona maps of arbitrarily large degree in positive characteristic

A result of I.V.Dolgachev states that the complex homaloidal polynomials in three variables, i.e. the complex homogeneous polynomials whose polar map is birational, are of degree at most three. In this note we describe homaloidal polynomials in three variables of arbitrarily large degree in positive characteristic. Using combinatorial arguments, we also classify line arrangements whose polar map is homaloidal in positive characteristic.