Page 124 - Geometric Modeling and Algebraic Geometry

7 Curve Parametrization Exploiting the Newton Polygon 125
• S is smooth, in fact it is covered by affine planes. Hence we are locally working
with polynomials.
• S is a complete algebraic variety (see for example [3] or [5]).
∗
• S contains the torus T =(K ) as a dense open subset.
2
7.2.4 Completion of the curve

The parametrization problem for curves is a problem of birational geometry. To solve
it, we have to apply certain theorems of “global content”. Therefore we have to study
a complete model of our curve, that is a curve without any missing points.
The Newton polygon Π(f) ⊂ R is defined as the convex hull of all lattice points
2
r s
(r, s) ∈ Supp(f) (i.e. all (r, s) ∈ Z s.t. x y appears with a non-zero coefficient
2
in f). An absolutely irreducible polynomial f ∈ K[x, y] defines an irreducible curve
in the affine plane A .If Π(f) is non-degenerate, i.e. has dimension 2, then f also
2
K
defines a curve on the torus T ⊂ A .
2
K
From now on we also fix the surface S which is constructed from Π(f) as in
the previous section. S contains the torus and hence we can define C to be the
Zariski closure of the curve defined by f on the torus. If the half planes used for
the construction of S are determined by the integers a i ,b i ,c i then C is defined by
the polynomials
v
f i (u i ,v i ):= u −c i−1 −c i f(u a i−1 a i b i−1 b i (7.4)
v ,u
v )
i i i i i i
within the open subsets U i ⊂ S. As a closed subset of a complete space, C is com-
plete itself.
For example if f is a dense polynomial with respect to total degree, meaning that
it contains all monomials up to a certain degree, then Π(f) is a triangle. If f is a
dense polynomial with respect to bidegree, then Π(f) is a rectangle. So in the first
case we would work inside P , in the second case inside P × P . In general the
1
1
2
K K K
surface is adapted to the Newton polygon, which is of course a much finer shape
parameter than any notion of degree.
We consider a polynomial to be sparse, if the shape of its Newton polygon differs
from an isosceles triangle. In this case our algorithm is more efficient than algorithms
relying on a projective embedding.
Throughout this article we will always implicitly assume that f is absolutely
irreducible and Π(f) is non-degenerate. For parametrizing in the other cases it is
easy to devise specialized algorithm, see also [1].

7.3 Divisors

In this section we introduce divisors and linear systems associated to them.