  We give lower bounds for the degree of the discriminant with respect to y of
separable polynomials f in K[x,y] over an algebraically closed field of
characteristic zero. Depending on the invariants involved in the lower bound,
we give a geometrical characterisation of those polynomials having minimal
discriminant, and give an explicit construction of all such polynomials in many
cases. In particular, we show that irreducible monic polynomials with minimal
discriminant coincide with coordinate polynomials. We obtain analogous partial
results for the case of nonmonic or reducible polynomials by studying their
GL2(K[x])-orbit and by establishing some combinatorial constraints on their
Newton polytope. Our results suggest some natural extensions of the embedding
line theorem of Abhyankar-Moh and of the Nagata-Coolidge problem to the case of
unicuspidal curves of P1 x P1.
