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### Computing the places of

A place of is represented by a triple, consisting of

• the closed point corresponding to ,
• the degree of the place (that is, the minimal degree of a field extension defining ) and
• a symbolic Hamburger-Noether expression HN for the local branch corresponding to (defined over a primitive algebraic field extension of degree ).
Recall that denotes the formal sum of a point with its conjugates. Affine closed points will be represented by a defining (triangular) ideal , while closed points at infinity are usually stored in form of a homogeneous polynomial (the defining prime factor of ).

Note that the conjugates of a place are given by the triples HN, where HN runs through the conjugates of HN. Hence, when computing the closed places of , we can restrict ourselves to computing one representing place for each. We apply the following algorithm:

Input
Squarefree homogeneous polynomial ,
degree bound .

Output
List of all closed singular places and all closed non-singular places up to degree of the plane curve defined by .

1. Affine singular points. Let and the Tjurina ideal of . Compute a triangular system for , that is, a system of triangular bases such that .

Here, by a triangular basis one denotes a reduced lexicographical Gröbner basis of the form with a monic polynomial in and . Triangular systems can be computed effectively, basically by two different methods, one due to Lazard [27,7], the other due to Möller [31]. Choose any of these methods to compute a triangular system for , , . For each ,

• compute a prime factorization of in ,

• for , let be the primitive field extension defined by the irreducible polynomial . Compute a prime factorization of in ,

Finally, the closed affine singular points are given by the set of ideals

where is the image of when substituting the parameter by .

2. Points at infinity. Let and compute a prime factorization of the polynomial ,

 (4)

Let be a root of and define

where denotes the formal sum of the point (defined over ) with its conjugates. (It is represented by .)

We denote by the subset of closed singular points. To check whether a point is singular or not, one has to check whether (these computations can be performed over the finite field extension ).

Finally, consider the (closed) point : if then has to be added to ; if, additionally, and vanish then it has to be added to , too.

The sets (resp. ) are the sets of closed (singular) points at infinity.

3. Affine singular places. To each closed affine singular point given by a (triangular) ideal we compute the corresponding places in form of a system of symbolic Hamburger-Noether expressions for the respective germ (defined over ). More precisely, a closed place over is the formal sum of a place described by one of the computed sHNE with its conjugates.

The computation of the symbolic Hamburger-Noether expressions has to be performed in the local ring where is a primitive field extension (of degree ) such that decompose into linear factors. Note that during the computation of a sHNE further field extensions might be necessary.

4. Singular places at infinity. To each closed singular point in we compute a system of sHNE for the local germ (defined over ). To be precise, if then we compute a system of sHNE for in ; if then the system of sHNE is computed for in .

5. Non-singular affine closed points up to degree . For each do the following:
• let and set .
• Proceed as in Step 1 to obtain a set of (triangular) ideals corresponding to the set of closed points defined over .
• For all non-singular (given by ) compute the degree ). If then compute the corresponding closed place (that is, a sHNE for the germ ) and add it to the list of closed places.

Remark 2.10   It is interesting to notice that triangular sets have mainly been used for numerical purpose, since they allow a fast and stable numerical solving of polynomial systems (cf. [27,31,13]), and this has been the reason for implementing it in SINGULAR. Several experiments have shown that they behave also superior against other methods to represent closed points over finite fields.

Next: Computing the adjunction divisor Up: Computational Aspects Previous: Computational Aspects
Christoph Lossen
2001-03-21