I am reading the paper [A Gröbner fan method for biochemical network modeling](https://dl.dropboxusercontent.com/u/6980502/Uni/Paper-1-Gr%C3%B6bner-Fan-Method.pdf).


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?



Vanishing ideal of a finite set of points does not have expected amount of cones in Gröbner fan
Thank you for your question. I am one of the authors of the paper. 


To be clear, the points from which the ideal of points is created are the inputs (not the outputs) of the data. What you have done seems correct: you computed the ideal of the input points over ZZ/2 and then computed the Groebner fan of the ideal. I recomputed the Groebner fan using Macaulay2 and found 3 cones, just as you found. 


At this point, I cannot resolve the discrepancy. If I do, I will post a subsequent response. 


