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 ).

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 .

*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
,

- compute a prime factorization of
in
,
*Points at infinity.*Let and compute a prime factorization of the polynomial ,

Let be a root of and defineWe 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.

*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.

*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 .

*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.