Page 132 - Geometric Modeling and Algebraic Geometry

7 Curve Parametrization Exploiting the Newton Polygon 133
7.6 The algorithm

We now summarize the resulting algorithm. For the input equation f ∈ K[x, y] of
our algorithm we require:
Condition (*): The Newton polygon Π(f) is non-degenerate and we can find
l 1 ,k 1 ,l 2 ,k 2 as in section 7.5.4.
Condition (*) is fulfilled if Π(f) is non-degenerate and all edges have length
≥ 2. If there is an edge with length =1 then the algorithm of paper [1] can be used
to compute a rational parametrization without field extension, because in this case it
is easy to find a rational point on the orignal curve. Therefore (*) is not critical.
Some remarks to algorithm 1 are in order. The delta invariants δ P can be com-
puted using Puiseux expansions or Hamburger-Noether expansions (in the case of
positive characteristic), see [6]. Implementations exist in Maple (characteristic zero
only) and in Singular [7]. These tools also provide a way to represent the valuations
ν by suitable homomorphisms into Laurent series rings.

7.7 Example

We want to parametrize the curve C ⊂ A given implicitly by the equation
2
Q
f = −27y x − 4y x +13y x +8yx − 20y x − 8yx 2
2 2
3
2 3
2
3
+4y − 8yx +4x +8y +8x +4.
2
2
Hence the field of definition is Q. The Newton polygon Π(f) is depicted in figure
7.4. It has 6 vertices and 4 interior points. It can be represented as the intersection of
8 half planes which are governed by the following set of data:
v 1 =(0, 0), (a 1 ,b 1 )=(0, 1), c 1 =0,
v 2 =(2, 0), (a 2 ,b 2 )=(−1, 1), c 2 = −2,
v 3 =(3, 1), (a 3 ,b 3 )=(−1, 0), c 3 = −3,
v 4 =(3, 2), (a 4 ,b 4 )=(−1, −1), c 4 = −5,
v 5 =(3, 2), (a 5 ,b 5 )=(−1, −2), c 5 = −7,
v 6 =(1, 3), (a 6 ,b 6 )=(0, −1), c 6 = −3,
v 7 =(1, 3), (a 7 ,b 7 )=(1, −1), c 7 = −2,
v 8 =(0, 2), (a 8 ,b 8 )=(1, 0), c 8 =0

The half planes with normals (a 4 ,b 4 ) and (a 6 ,b 6 ) have been inserted in order to
fulfill equation (7.3). Hence the constructed toric surface is covered by 8 affine charts

S = U i . Using (7.4) we compute the equations in all the charts:
i∈[1,8]