Singular

Hi, all,

I found an tricky problem:
> ring r=(integer, 2, 3), (a,b,c,d,e,f,s),dp;
> ideal i=f-a*b-a*c,e-a*d,d-b-c,s*(f-e)-4;
> option(redSB);
> std(i);
_[1]=4
_[2]=e-f
_[3]=b+c-d
_[4]=ad-f

Have a look at the result, it is not a reduced minimal Groebner Basis!
Because according to the definition of minimal Groebner Basis, all the coefficients of generators in GB should be 1.
Here _[1]=4 which is not 1.

Any comments?

Thanks

Dear gepo,
this does only apply to Gröbner bases of polynomials whose coefficient domain is a field.
In your example, Z/8 (this is not the Galois field with 8 elements) is used, which is a proper ring (no field).

Regards,
Alexander

Dear Alexander,
Could you tell a little bit more about what the conditions are for a minimal reduced Groebner Basis over ring?

I am not quite clear about that.

Thanks a lot.

Dear gepo,
we call G a reduced Gröbner basis, if no term from tail(g) for any g ∈ G is divisible by a leading term of an element of G.
Note also, that you cannot have normalized Gröbner bases in every case. For instance, the ideal <2*x> over Z cannot be simplified into <x>, because Z has no multiplicative inverse for non-units.

For a brief description of Gröbner Bases over rings chapter 2 of
http://dx.doi.org/10.1016/j.jpaa.2008.11.043

Best regards,
Alexander