Computing the Newton Polygon of the Implicit Equation

We consider rationally parameterized plane curves, where the polynomials in the parameterization have fixed supports and generic coefficients. We apply sparse (or toric) elimination theory in order to determine the vertex representation of the implicit equation’s Newton polygon. In particular, we consider mixed subdivisions of the input Newton polygons and regular triangulations of point sets defined by Cayley’s trick. We consider polynomial and rational parameterizations, where the latter may have the same or different denominators; the implicit polygon is shown to have, respectively, up to 4, 5, or 6 vertices. Mathematics Subject Classification (2000). Primary 14H50; Secondary 14M25, 14Q05, 52B20.