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### Stanley-Reisner ideal from polytope

Hi!

I'm just discovering how sage works and i need to get Stanley-Reisner ideal from polytope, knowing coordinates of verticies. Is there a way to do it? 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

Danilovâ€“Jurkiewicz theorem. See for example [https://www.mathnet.ru/php/getFT.phtml?jrnid=rm&paperid=9883&what=fullteng&option_lang=rus, 8.6]

I got that data:

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 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:

$p2= new Polytope(POINTS=>[[1, 1,0,0,0],[1, -1,0,0,0],[1, 0,1,0,0],[1, 0,0,1,0],[1, 0,0,0,1],[1, 0,-1,-1,-1]]);$HD2 = $p2->HASSE_DIAGRAM; print$HD2->FACES;

Then i figure out what sets are missing and find minimal of them. But at some point i got mistake, thats why i need some other way to calucalte that ideal.

I tried something like that:

P = Polyhedron(vertices=[[1,0,0,0],[-1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1],[0,-1,-1,-1]])
I = P.stanley_reisner_ideal()


But it is not working. As i can understand it's can be done in some more complicated way, that i cant discover. How can i do this?