The 𝐷-resultant, singularities and the degree of unfaithfulness

Abstract

We introduce the D 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 ] 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 ) 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.