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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:

sum of \lambda_i*v_i

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

Danilov–Jurkiewicz theorem. See for example [https://www.mathnet.ru/php/getFT.phtml? ... n_lang=rus, 8.6]

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

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:

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;

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:

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

Or my method is wrong?

Posted: **13 Jun 2024, 12:32**

Your approach only works in the smooth case. In that case the toric variety (defined by the normal fan of the polytope) is simplicial. Intersecting the normal fan gives a finite simplicial complex, and its non-faces give you the Stanley-Reisner ring. See polymake's topaz documentation. See also OSCAR.

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Stanley-Reisner ideal of polytope

Stanley-Reisner ideal of polytope

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