I am reading the paper A Gröbner fan method for biochemical network modeling.
In Chapter 4.3 (i.stack.imgur.com/h2O8B.png) they calculate the vanishing ideal of some tuples (input points of Series 1-4) in the integer ring modulo $2$.
They say that this ideal should have a Gröbner fan with $13$ cones, but I cannot reproduce that.
What I did
I determined the vanishing ideal using maple (Did I use the correct points?):
> with(PolynomialIdeals):
> VanishingIdeal([[0, 1, 0, 0, 0], [0, 0, 0, 0, 0], [1, 1, 0, 0, 0], [0, 1, 0, 0, 1],
[0, 0, 0, 1, 0], [1, 0, 1, 1, 0], [0, 1, 0, 1, 0]], [a, b, c, d, e], 2);
a*e, c*b, c*e, e*d, a^2+a, b^2+b, c^2+c, d^2+d, e^2+e, a*c+c, b*e+e, c*d+c, a*d+c,
a*b+a+c
If I format this correctly for the tool gfan and run it, I get a Gröbner fan with $3$ cones instead of $13$.
$ gfan
Z/2Z[a,b,c,d,e]
{ae,cb,ce,ed,aa+a,bb+b,cc+c,dd+d,ee+e,ac+c,be+e,cd+c,ad+c,ab+a+c}
Z/2Z[a,b,c,d,e]
{{
e^2+e,
d*e,
d^2+d,
c*e,
c*d+c,
c^2+c,
b*e+e,
b*c,
b^2+b,
a*e,
a*d+c,
a*c+c,
a*b+c+a,
a^2+a}
,
{
e^2+e,
d*e,
d^2+d,
c+a*d,
b*e+e,
b^2+b,
a*e,
a*b+a+a*d,
a^2+a}
,
{
e^2+e,
d*e,
d^2+d,
c+a+a*b,
b*e+e,
b^2+b,
a*e,
a*d+a+a*b,
a^2+a}
}
For gfan see: home.math.au.dk/jensen/software/gfan/gfan.html
Does anybody have an idea what went wrong?
Maybe I did not use the correct points for computing the ideal?