### Stanley-Reisner ideal of polytope

Posted:

**12 Jun 2024, 03:45**Hi!

I want to find Stanley-Reisner ideal of polytope, knowing coordinates of verticies. It is defined in such way:

Stanley-Reisner ideal is generated by two relations:

I got that data:

I know that first type relation i get will look like this:

v0-v1

v1-v3

v2-v5

v3-v5

v4-v5

But for second one i need more calculations, because i can't tell what faces are intersecting knowing only verticies (i need to calculate it hundreds of times, so thats why i need computer).

I did some calculations in polymake:

Then i figure out what sets are missing and find minimal of them. But at some point i got mistake. Do someone know the reason it can be? Maybe there are some restrictions on dimension of polytope or number of it's verticies? First mistake is founded on 4-dim polytope with 8 verticies:

Or my method is wrong?

I want to find Stanley-Reisner ideal of polytope, knowing coordinates of verticies. It is defined in such way:

Stanley-Reisner ideal is generated by two relations:

- sum of \lambda_i*v_i

- multiplication of v_i1,...,v_ik if intersection F_i1,...,F_ik is empty

I got that data:

Code: Select all

```
dim=4, vert=6
1 -1 0 0 0 0
0 0 1 0 0 -1
0 0 0 1 0 -1
0 0 0 0 1 -1
```

v0-v1

v1-v3

v2-v5

v3-v5

v4-v5

But for second one i need more calculations, because i can't tell what faces are intersecting knowing only verticies (i need to calculate it hundreds of times, so thats why i need computer).

I did some calculations in polymake:

Code: Select all

```
p2=newPolytope(POINTS=>[[1,1,0,0,0],[1,−1,0,0,0],[1,0,1,0,0],[1,0,0,1,0],[1,0,0,0,1],[1,0,−1,−1,−1]]);
HD2 = p2−>HASSEDIAGRAM;print
HD2->FACES;
```

Code: Select all

```
dim=4, vert=8
1 0 0 0 0 -1 1 0
0 1 0 -1 0 0 1 0
0 0 1 -1 0 0 1 -1
0 0 0 0 1 -1 0 1
```