  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.
