Reduced Groebner Basis

rambiz · Thu Jun 09, 2016 4:44 pm

Hi all;
the coefficients of the leading terms of the generators of a reduced Groebner basis should be equal to one (1) as far as I know. Now consider the following code:

Code:

> ring r=0,(x,y),lp;
> ideal i=x4+x2+xy3+2,x2+y2-1;
> option(redSB);
> ideal g=std(i);
> g;
g[1]=2y8-7y6+17y4-24y2+16
g[2]=16x-6y7+13y5-23y3+20y
>

Here the leading terms have the coefficients 2 and 16. How comes? What am I missing?

Re: Reduced Groebner Basis

hannes · Fri Jun 10, 2016 11:00 am

No, that depends on the definition.
Any (skalar) multiples of the elements of a Groebner basis form also
also a Groebner basis.
(see http://www.singular.uni-kl.de/Manual/4-0-3/sing_900.htm)
Some authors divied all elements by the leading coeficient
(and get then a unique Groebner basis, if it is completely reduced)

Re: Reduced Groebner Basis

rambiz · Sat Jun 11, 2016 4:20 pm

Thank you for the clarification. For sure it is a matter of definition! Nonetheless I think a few words about the definition of the Reduced Groebner Basis used by the Singular developers would be helpful. It can definitely confuse a few new users.