Plane curves with minimal discriminant

Abstract

We give lower bounds for the degree of the discriminant with respect to $y$ of squarefree polynomials $f\in \mathbb {K}[x,y]$ over an algebraically closed field of characteristic zero. Depending on the invariants involved in the lower bound, we give a geometrical characterization of those polynomials having minimal discriminant, and we 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 $GL_2(\mathbb {K}[x])$-orbit and by establishing some combinatorial constraints on their Newton polygon. 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 $\mathbb {P}^1\times \mathbb {P}^1$.