# simplex -- Simplex in the variables of a polynomial ring.

## Synopsis

• Usage:
simplex(R)
simplex(R,Rdual)
• Inputs:
• R, ,
• Rdual, ,
• Optional inputs:
• computeFaces => ..., default value null, Compute faces of a simplex.
• Outputs:
• ,

## Description

Returns a simplex on the variables of R.

The Option computeFaces=>false suppresses the computation of all faces.

If Rdual is specified it is used for the vertices of the dual simplex, if not a new polynomial ring is created. It is graded by the coordinates of the vertices of the dual simplex.

The dual simplex is always created without face data.

 i1 : R=QQ[x_0..x_4] o1 = R o1 : PolynomialRing i2 : C=simplex(R) o2 = 4: x x x x x 0 1 2 3 4 o2 : complex of dim 4 embedded in dim 4 (printing facets) equidimensional, simplicial, F-vector {1, 5, 10, 10, 5, 1}, Euler = 0 i3 : grading C o3 = | -1 -1 -1 -1 | | 1 0 0 0 | | 0 1 0 0 | | 0 0 1 0 | | 0 0 0 1 | 5 4 o3 : Matrix ZZ <--- ZZ i4 : dC=C.dualComplex o4 = 4: v v v v v 0 1 2 3 4 o4 : complex of dim 4 embedded in dim 4 (printing facets) equidimensional, simplicial i5 : grading dC o5 = | -1 -1 -1 4 | | -1 -1 4 -1 | | -1 4 -1 -1 | | 4 -1 -1 -1 | | -1 -1 -1 -1 | 5 4 o5 : Matrix QQ <--- QQ i6 : fc(dC); i7 : dC o7 = 4: v v v v v 0 1 2 3 4 o7 : complex of dim 4 embedded in dim 4 (printing facets) equidimensional, simplicial, F-vector {1, 5, 10, 10, 5, 1}, Euler = 0