# [GAP Forum] How to compute a simplified Groebner basis

Jason ljshan at gmail.com
Thu Aug 13 08:00:23 BST 2009

```Hello,
I am a first time user in GAP, if the following question is dummy,

I tried to use Buchberger's Algorithm to compute a Groebner basis for an
ideal of a polynomial ring in rational field, here is my GAP script:

F := Rationals;
R := PolynomialRing( F, [ "x", "y", "z" ]);
x := IndeterminatesOfPolynomialRing(R)[1];
y := IndeterminatesOfPolynomialRing(R)[2];
z := IndeterminatesOfPolynomialRing(R)[3];
I := Ideal (R, [x^2+2*x*y^2, x*y + 2*y^3 - 1]);
ord := MonomialLexOrdering(x,y,z);

GroebnerBasis( I, ord );

but the result is always this:

[ 2*x*y^2+x^2, 2*y^3+x*y-1, -x, -4*y^4+2*y, 2*y^3-1 ]

Obviously, the fourth can be completely divided by the last basis, the
first and second can be completely divided by the third basis. The
expected result should be like this:

[ -x, 2*y^3-1 ]

So my question is how to get the simplified Groebner basis in GAP ?

Here is my GAP info:
GAP4, Version: 4.4.10 of 02-Oct-2007,
i486-pc-linux-gnu-i486-linux-gnu-gcc

regards,
jason

```