Enumeration of Places

In several situations one needs to loop over the places of a function field until either the one finds a place with special properties or until they generate a certain group. The functions listed here support this.

PlaceEnumInit(K): FldFun -> PlcEnum

Coprime: Any                        Default: 
All    : BoolElt                    Default: false

Initialises an enumeration process for places of the function field \(K\). The enumeration process will loop over all irreducible polynomials of the underlying finite field and for each polynomial over all primes lying above it.

If Coprime is given, it should be either a set of places that should be ignored in the process or a divisor. In case a divisor is passed in only places coprime to the divisor will be returned.

If All is false, the infinite places won’t be considered.

PlaceEnumInit(P): PlcFunElt -> PlcEnum

Coprime: Any                    Default: 

Constructs an enumeration process for places starting at the place \(P\).

If Coprime is given, it should be either a set of places that should be ignored in the process or a divisor. In case a divisor is passed in only places coprime to the divisor will be returned.

PlaceEnumInit(K, Pos): FldFun, [RngIntElt] -> PlcEnum

Coprime: Any                    Default: 

Constructs an enumeration environment that starts at the place of the function field \(K\) indexed by Pos as returned from PlaceEnumPosition.

PlaceEnumCopy(R): PlcEnum -> PlcEnum

Copies the environment and the current state of the enumeration process for places \(R\).

PlaceEnumPosition(R): PlcEnum -> [RngIntElt]

Returns a list of integers that acts as an index to the places as enumerated by the environment \(R\).

PlaceEnumNext(R): PlcEnum -> PlcFunElt

Returns the “next” place of the process \(R\).

PlaceEnumCurrent(R): PlcEnum -> PlcFunElt

Returns the current place pointed to by the environment \(R\), i.e. the last place returned by PlaceEnumNext.