This method returns the ring of continuous piecewise polynomials on the complex $\Delta$ with vertices $V$ and facets $F$. If $\Delta$ is a simplicial complex, write $\Delta_0$ for its vertices and $K[\Delta]$ for the Stanley-Reisner ring of $\Delta$, which is the quotient of the polynomial ring $K[X_v:v\in\Delta_0]$ by monomials corresponding to non-faces. Then, as a ring, $C^0(\Delta)$ is isomorphic to the so-called affine Stanley Reisner ring $K_a[\Delta]=\frac{K[\Delta]}{(1-\sum_{v} X_v)K[\Delta]}$. If $\Delta$ is a cone, then $K_a[\Delta]=K[\Delta_{dc}]$, where $\Delta_{dc}$ is the decone of $\Delta$. The map $K_a[\Delta]\rightarrow C^0(\Delta)$ is given by $X_v\rightarrow C_v$, where $C_v$ is the Courant function at the vertex $v$. See courantFunctions.
i1 : V={{0,0},{0,1},{-1,-1},{1,0}};
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i2 : F={{0,1,2},{0,2,3},{0,1,3}};
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i3 : phi=stanleyReisner(V,F)
QQ[e ..e , t ..t ]
0 2 0 2
o3 = map (---------------------------------------------, QQ[X ..X ], {2e t - e t + e t - e t + 2e t + e t - e t - e t + e t , - e t + e t + e t , - e t - e t , e t - e t + e t })
2 2 2 0 3 0 0 0 1 0 2 1 0 1 1 1 2 2 0 2 1 2 2 0 0 0 1 2 1 0 0 1 1 1 0 1 1 2 0
(e e , e e , e e , e - e , e - e , e - e )
0 1 0 2 1 2 0 0 1 1 2 2
QQ[e ..e , t ..t ]
0 2 0 2
o3 : RingMap --------------------------------------------- <--- QQ[X ..X ]
2 2 2 0 3
(e e , e e , e e , e - e , e - e , e - e )
0 1 0 2 1 2 0 0 1 1 2 2
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i4 : ker phi--decone of homogenized simplicial complex is three triangles meeting at a vertex
o4 = ideal(X X X )
1 2 3
o4 : Ideal of QQ[X ..X ]
0 3
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i5 : R=QQ[x,y]; |
i6 : phi=stanleyReisner(V,F,BaseRing=>R,Homogenize=>false)
QQ[e ..e , x..y]
0 2
o6 = map (---------------------------------------------, QQ[X ..X ], {- e x + e y + e y, - e x - e y, e x - e y + e x})
2 2 2 0 2 0 0 2 0 1 1 1 2
(e e , e e , e e , e - e , e - e , e - e )
0 1 0 2 1 2 0 0 1 1 2 2
QQ[e ..e , x..y]
0 2
o6 : RingMap --------------------------------------------- <--- QQ[X ..X ]
2 2 2 0 2
(e e , e e , e e , e - e , e - e , e - e )
0 1 0 2 1 2 0 0 1 1 2 2
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i7 : ker phi--decone of simplicial complex is a three-cycle
o7 = ideal(X X X )
0 1 2
o7 : Ideal of QQ[X ..X ]
0 2
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The Stanley Reisner ring of a simpicial complex is obtained using the function ringStructure, where the ring generators are chosen to be the Courant functions.
i8 : V={{0,0},{0,1},{-1,-1},{1,0}};
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i9 : F={{0,1,2},{0,2,3},{0,1,3}};
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i10 : R=QQ[x,y]; |
i11 : CF = courantFunctions(V,F,Homogenize=>false,BaseRing=>R);
3 3
o11 : Matrix R <--- R
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i12 : phi=ringStructure(image CF,VariableGens=>false)
QQ[e ..e , x..y]
0 2
o12 = map (---------------------------------------------, QQ[Y ..Y ], {- e x + e y + e y, - e x - e y, e x - e y + e x})
2 2 2 0 2 0 0 2 0 1 1 1 2
(e e , e e , e e , e - e , e - e , e - e )
0 1 0 2 1 2 0 0 1 1 2 2
QQ[e ..e , x..y]
0 2
o12 : RingMap --------------------------------------------- <--- QQ[Y ..Y ]
2 2 2 0 2
(e e , e e , e e , e - e , e - e , e - e )
0 1 0 2 1 2 0 0 1 1 2 2
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i13 : ker phi
o13 = ideal(Y Y Y )
0 1 2
o13 : Ideal of QQ[Y ..Y ]
0 2
|
If $\Delta$ is not simplicial, then the image of the map is a minimal set of generators for the ring $C^0(\Delta)$. There is no canonical choice of generators, and $C^0(\Delta)$ is no longer a combinatorial object.
i14 : V={{0,1},{-1,-1},{1,-1},{0,10},{-2,-2},{2,-2}};--symmetric triangular prism
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i15 : V'={{0,1},{-1,-1},{1,-1},{1,10},{-2,-2},{2,-2}};--asymmetric triangular prism
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i16 : F={{0,1,2},{0,1,3,4},{0,2,3,5},{1,2,4,5}};
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i17 : S=QQ[x,y,z]; |
i18 : phi=stanleyReisner(V,F,BaseRing=>S) --four generators in degree one
QQ[e ..e , x..z]
0 3
o18 = map (------------------------------------------------------------------------, QQ[X ..X ], {e x + e x + e x + e x, e y + e y + e y + e y, e z + e z + e z + e z, e y + e z - 2e x + 2e y + 2e x + 2e y})
2 2 2 2 0 3 0 1 2 3 0 1 2 3 0 1 2 3 0 0 1 1 2 2
(e e , e e , e e , e e , e e , e e , e - e , e - e , e - e , e - e )
0 1 0 2 1 2 0 3 1 3 2 3 0 0 1 1 2 2 3 3
QQ[e ..e , x..z]
0 3
o18 : RingMap ------------------------------------------------------------------------ <--- QQ[X ..X ]
2 2 2 2 0 3
(e e , e e , e e , e e , e e , e e , e - e , e - e , e - e , e - e )
0 1 0 2 1 2 0 3 1 3 2 3 0 0 1 1 2 2 3 3
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i19 : phi'=stanleyReisner(V',F,BaseRing=>S) --six generators in degrees one and two
QQ[e ..e , x..z]
0 3 2 2 2 2 2 2 2 2 2 2 2 2
o19 = map (------------------------------------------------------------------------, QQ[X ..X ], {e x + e x + e x + e x, e y + e y + e y + e y, e z + e z + e z + e z, - 14e x*y - 7e y - 14e x*z + 7e z + 72e x - 80e x*y + 8e y + 8e x*z - 8e y*z, 2e x*y + 9e y + 2e x*z + 9e y*z - 20e x*y + 20e y + 2e x*z - 2e y*z + 16e x*y + 16e y + 2e x*z + 2e y*z, 14e x*y - 47e y + 14e x*z - 36e y*z + 11e z + 80e x*y - 80e y - 8e x*z + 8e y*z + 72e x - 72e y })
2 2 2 2 0 5 0 1 2 3 0 1 2 3 0 1 2 3 0 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 2 2 2 2 0 0 0 0 0 1 1 1 1 2 2
(e e , e e , e e , e e , e e , e e , e - e , e - e , e - e , e - e )
0 1 0 2 1 2 0 3 1 3 2 3 0 0 1 1 2 2 3 3
QQ[e ..e , x..z]
0 3
o19 : RingMap ------------------------------------------------------------------------ <--- QQ[X ..X ]
2 2 2 2 0 5
(e e , e e , e e , e e , e e , e e , e - e , e - e , e - e , e - e )
0 1 0 2 1 2 0 3 1 3 2 3 0 0 1 1 2 2 3 3
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i20 : ker phi --kernel generated by single polynomial of degree four
2 3 2 2 2 2 2 2 2
o20 = ideal(4X X X - 4X X + 4X X X - 4X X X - 4X X + 8X X + 4X X X -
0 1 3 1 3 0 2 3 1 2 3 0 3 1 3 1 2 3
-----------------------------------------------------------------------
3 3 4
5X X - X X + X )
1 3 2 3 3
o20 : Ideal of QQ[X ..X ]
0 3
|
i21 : mingens ker phi' --kernel has six minimal generators of degree four
o21 = | 76X_0X_1X_4-428X_1^2X_4-22X_0X_2X_4-22X_1X_2X_4+11X_4^2-16X_0X_1X_5-
-----------------------------------------------------------------------
72X_1^2X_5-2X_0X_2X_5-9X_1X_2X_5+X_4X_5
-----------------------------------------------------------------------
67032X_0^2X_4-1663704X_1^2X_4-99792X_0X_2X_4-99792X_1X_2X_4-2299X_3X_4+
-----------------------------------------------------------------------
49896X_4^2-9576X_0^2X_5-110880X_0X_1X_5-278712X_1^2X_5-13860X_0X_2X_5-
-----------------------------------------------------------------------
36036X_1X_2X_5-209X_3X_5+5999X_4X_5+133X_5^2
-----------------------------------------------------------------------
14630X_1X_2X_3-1463X_2^2X_3+145152X_1^2X_4+61740X_0X_2X_4-43596X_1X_2X_
-----------------------------------------------------------------------
4+1368X_3X_4-4536X_4^2+10080X_0X_1X_5+26208X_1^2X_5+1260X_0X_2X_5-8428X
-----------------------------------------------------------------------
_1X_2X_5-1463X_2^2X_5+342X_3X_5-630X_4X_5
-----------------------------------------------------------------------
146300X_1^2X_3-1463X_2^2X_3+1132992X_1^2X_4+123480X_0X_2X_4+18144X_1X_
-----------------------------------------------------------------------
2X_4+1368X_3X_4-35406X_4^2+20160X_0X_1X_5+146048X_1^2X_5+2520X_0X_2X_5+
-----------------------------------------------------------------------
6552X_1X_2X_5-1463X_2^2X_5+342X_3X_5-1260X_4X_5
-----------------------------------------------------------------------
4180X_0X_1X_3-418X_0X_2X_3+116928X_1^2X_4+7308X_0X_2X_4+7308X_1X_2X_4+
-----------------------------------------------------------------------
209X_3X_4-3654X_4^2+6448X_0X_1X_5+19440X_1^2X_5+806X_0X_2X_5+2430X_1X_
-----------------------------------------------------------------------
2X_5-403X_4X_5 2633400X_0^2X_3-26334X_2^2X_3-36575X_3^2+281667456X_1^2X
-----------------------------------------------------------------------
_4+18552240X_0X_2X_4+16656192X_1X_2X_4+640224X_3X_4-8802108X_4^2+
-----------------------------------------------------------------------
2633400X_0^2X_5+18506880X_0X_1X_5+47169864X_1^2X_5+2313360X_0X_2X_5+
-----------------------------------------------------------------------
6014736X_1X_2X_5-26334X_2^2X_5+86906X_3X_5-1156680X_4X_5-36575X_5^2 |
1 6
o21 : Matrix (QQ[X ..X ]) <--- (QQ[X ..X ])
0 5 0 5
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The object stanleyReisner is a method function with options.