Plane curves with minimal discriminant

Martin Weimann (1, 2, 3) , Denis Simon (1, 2)

1 UNICAEN - Université de Caen Normandie
2 LMNO - Laboratoire de Mathématiques Nicolas Oresme
3 GAATI - Laboratoire de Géométrie Algébrique et Applications à la Théorie de l'Information

We give lower bounds for the degree of the discriminant with respect to y of squarefree polynomials f ∈ 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 poly-nomials 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 GL 2 (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 P 1 × P 1 .

Domaines: Mathématiques [math]; Algèbre commutative [math.AC]; Géométrie algébrique [math.AG]

Martin Weimann, Denis Simon. Plane curves with minimal discriminant. Journal of Commutative Algebra, 2018, 10 (4), pp.559-598. ⟨10.1216/JCA-2018-10-4-559⟩. ⟨hal-02137328⟩