The Newton Polygon of Plane Curves with Many Rational Points

The Newton Polygon of a Rational Plane Curve

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 KušnirenkoBernštein theorem. We apply this result to the determination of the Newton polygon of a curve parameterized by generic Laurent polynomials or by generic r...

Sparse Parametrization of Plane Curves

We present a new method for the rational parametrization of plane algebraic curves. The algorithm exploits the shape of the Newton polygon of the defining implicit equation and is based on methods of toric geometry.

Non-hyperelliptic curves of genus three over finite fields of characteristic two

Let k be a finite field of even characteristic. We obtain in this paper a complete classification, up to k-isomorphism, of non singular quartic plane curves defined over k. We find explicit rational normal models and we give closed formulas for the total number of k-isomorphism classes. We deduce from these computations the number of k-rational points of the different strata by the Newton polyg...

Points at Rational Distance from the Vertices of a Unit Polygon

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 d...