## Stanley-Reisner ideal of polytope

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grkj
Posts: 2
Joined: 01 Jun 2024, 01:57

### Stanley-Reisner ideal of polytope

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:

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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:

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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:

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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?

joswig
Main Author
Posts: 288
Joined: 24 Dec 2010, 11:10

### Re: Stanley-Reisner ideal of polytope

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.