The Newton Polygon of a Rational Plane Curve

The Newton Polygon of Plane Curves with Many Rational Points

This curve has the points (1 : 0 : 0) and (0 : 1 : 0) at infinity over any field. The affine equation is XY +Y +X = 0. The origin is a point of this curve. If (x, y) ∈ F8 is a point of this curve with nonzero coordinates, then x = 1. So 0 = xy + y + x = xy + xy + x = x[(xy) + (xy) + 1]. Let t = xy. Then t + t+ 1 = 0. So the Klein quartic has 3.7 = 21 rational points over F8 with nonzero coordin...

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.

A Compactness Criterion for Real Plane Algebraic Curves

Two sets of conditions are presented for the compactness of a real plane algebraic curve, one sufficient and one necessary, in terms of the Newton polygon of the defining polynomial.

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...