# simplexRing -- The underlying polynomial ring of a simplicial complex, face, fan or cone.

## Synopsis

• Usage:
simplexRing(D)
simplexRing(F)
• Inputs:
• D, an object of class Complex
• F, an object of class Face
• Outputs:
• R,

## Description

Returns the PolynomialRing of a simplicial Complex or a Face of a simplicial complex. This is the ring with variables corresponding to the vertices of D or the underlying complex D of F. It is the Stanley-Reisner ring of the simplex into which D embeds. This simplex is complex (entries vars R)#0

 `i1 : K=QQ;` `i2 : R=K[x_0..x_4];` ```i3 : I=ideal(x_0*x_1,x_1*x_2,x_2*x_3,x_3*x_4,x_4*x_0) o3 = ideal (x x , x x , x x , x x , x x ) 0 1 1 2 2 3 3 4 0 4 o3 : Ideal of R``` ```i4 : D=idealToComplex I o4 = {x x , x x , x x , x x , x x } 2 4 0 3 0 2 1 3 1 4 o4 : complex with 5 facets on the vertices x x x x x 0 1 2 3 4``` ```i5 : simplexRing D o5 = R o5 : PolynomialRing``` ```i6 : fc=facets D o6 = {x x , x x , x x , x x , x x } 2 4 0 3 0 2 1 3 1 4 o6 : List``` ```i7 : simplexRing fc#0 o7 = R o7 : PolynomialRing``` ```i8 : S=complex {(entries vars R)#0} o8 = {{x , x , x , x , x }} 0 1 2 3 4 o8 : complex with 1 facets on the vertices x x x x x 0 1 2 3 4``` ```i9 : complexToIdeal S o9 = ideal () o9 : Ideal of R```