Page 126 - Geometric Modeling and Algebraic Geometry

7 Curve Parametrization Exploiting the Newton Polygon 127
A . In this case g ∈ K(S) can be written as a rational function in K(u i ,v i ) for all
2
K
local coordinates u i and v i . The zeroes and poles (and their multiplicities) can be
read easily from the (absolute) factorization of the reduced representation.
On a toric surface we have a set of distinguished prime divisors, namely the
edge curves E i , also called the toric invariant prime divisors. They are clearly K-
rational. The linear systems of toric invariant divisors can easily be described by
support conditions.

Lemma 2. Let D = −˜ i E i ∈ Div(S) and define the polygon Π :=
c
1≤i≤n
r s
{(r, s) ∈ R | a i r + b i s ≥ ˜ i }. Then L S (D)= x y | (r, s) ∈ Π ∩ Z .
c
2
2
1≤i≤n K
Proof. See [1, Corollary 8].
In particular L S (D) = ∅ if and only if Π = ∅ and the basis is obviously contained
in K(S).
Remark 3. In our algorithm we use spaces of polynomials as above that are supported
approximately on the Newton polygon of f. If we considered only the degree of
the defining equation, the linear systems from above would correspond to spaces of
polynomials supported approximately on an isosceles triangle containing the Newton
polygon. As mentioned before, we consider a polynomial to be sparse, if its Newton
polygon differs from such a triangle. Therefore for sparse polynomials these vector
spaces become smaller and our algorithm becomes more efficient.

7.3.3 Divisors on smooth curves

For a smooth curve C, prime divisors correspond to points on the curve. A rational

function g ∈ K(C) can be developed at any point P ∈ C as a Laurent series with

respect to a local parameter.
More precisely let P ∈ C be a point. Then the local ring at the point P is regular

and we can find an injective homomorphism O(C) P → K[[t]] to a power series

ring. This homomorphism induces a homomorphism ϕ P : K(C) → K((t)).We

d
may assume that ϕ P is primitive, i.e. img(ϕ P ) ⊂ K((t )) for any d> 1. Then
ν P : K(C) → Z : g → ord t (ϕ P (h)) is a discrete valuation of the function field

K(C) over K with center P (see e.g. [19]).

Now if ν P (g) > 0 then g has a zero at P with multiplicity ν P (g),if ν P (g) < 0
then g has a pole at P with multiplicity −ν P (g).
7.3.4 Divisors on singular curves
If C is a complete, singular curve we may consider a resolution of singularities
π : C → C. I.e. C is a complete, smooth curve and π is a regular, birational map.

Such a resolution is known to exist and actually one can take the normalization of C
(see [16, II.5.1]).