The D-resultant, singularities and the degree of unfaithfulness

We introduce the D-resultant of two polynomials in one variable and show how it can be used to decide if k(f(t), g(t)) = k(t), k[f(t), g(t)] = k[t] and to find the singularities of the curve x = f(t), y = g(t). The second criterion is used to give a very short proof of a special case of the epimorphism theorem of Abhyankar and Moh.