The Newton polygon of a rational plane curve

Carlos d'Andrea (1, 2, 3) , Martin Sombra (4)

1 Department of Mathematics [Berkeley]
2 Departamento de Matemática [Buenos Aires]
3 Departament d'Algebra i Geometria
4 IMB - Institut de Mathématiques de Bordeaux

The Newton polygon of the implicit equation of a rational plane curve is explicitly determined by the multiplicities of any of its parametrizations. We give an intersection-theoretical proof of this fact based on a refinement of the Kuˇsnirenko- Bernˇstein theorem. We apply this result to the determination of the Newton polygon of a curve parameterized by generic Laurent polynomials or by generic rational func- tions, with explicit genericity conditions. We also show that the variety of rational curves with given Newton polygon is unirational and we compute its dimension. As a consequence, we obtain that any convex lattice polygon with positive area is the Newton polygon of a rational plane curve.

Mots clés: Rational plane curve; parametrization; implicit equation; Newton polygon; mixed integral

Domaines: Théorie des nombres [math.NT]

Carlos d'Andrea, Martin Sombra. The Newton polygon of a rational plane curve. 2007. ⟨hal-00359281⟩