# Difference between revisions of "ApCoCoA-1:FGLM.FGLM"

## FGLM.FGLM

Performs a FGLM Groebner Basis conversion using ApCoCoAServer.

### Syntax

```FGLM(GBInput:LIST, M:MAT):LIST
FGLM(GBInput:LIST):LIST
```

### Description

Please note: The function(s) explained on this page is/are using the ApCoCoAServer. You will have to start the ApCoCoAServer in order to use it/them.

The function FGLM calls the ApCoCoAServer to perform a

FGLM Groebner Basis conversion. Please note that the ideal generated by

the given Groebner Basis must be zero-dimensional. The Groebner Basis contained in list GBInput will be converted into a Groebner Basis with respect to term ordering Ord(M), i.e. M must be a matrix specifying a term ordering. If the parameter M is not specified, ApCoCoA will assume M = Ord(). Please note that the resulting polynomials belong to a different ring than the ones in GBInput.

The return value will be the transformed Groebner basis polynomials.

• @param GBInput A Groebner basis of a zero-dimensional ideal.

• @return A Groebner basis of the ideal generated by the polynomials of GBInput. The polynomials of the new Groebner basis will belong to the polynomial ring with term ordering specified by M or Ord() in case M is not given.

The following parameter is optional.

• @param M A matrix representing a term ordering.

#### Example

```Use QQ[x, y, z], DegRevLex;
GBInput := [z^4 -3z^3 - 4yz + 2z^2 - y + 2z - 2, yz^2 + 2yz - 2z^2 + 1, y^2 - 2yz + z^2 - z, x + y - z];
M := LexMat(3);
GBNew := FGLM.FGLM(GBInput, M);
Use QQ[x, y, z], Ord(M);
-- New basis (Lex)
BringIn(GBNew);

-------------------------------
[z^6 - z^5 - 4z^4 - 2z^3 + 1, y - 4/7z^5 + 5/7z^4 + 13/7z^3 + 10/7z^2 - 6/7z - 2/7,
x + 4/7z^5 - 5/7z^4 - 13/7z^3 - 10/7z^2 - 1/7z + 2/7]
-------------------------------
```