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# stanleyReisner -- Creates a ring map whose image is the ring of piecewise continuous polynomials on $\Delta$. If $\Delta$ is simplicial, the Stanley Reisner ring of $\Delta$ is returned.

## Synopsis

• Usage:
phi=stanleyReisner(V,F)
• Inputs:
• V, a list, a list of vertex coordinates
• F, a list, a list of facets, recorded as indices of V
• Optional inputs:
• BaseRing (missing documentation) => a ring, default value null,
• Homogenize (missing documentation) => , default value true,
• CoefficientRing (missing documentation) => a ring, default value QQ,
• VariableName (missing documentation) => , default value t,
• InputType (missing documentation) => , default value "ByFacets",
• Outputs:
• phi, , a ring map whose image is the ring of piecewise polynomials on $\Delta$.

## Description

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}}; i2 : F={{0,1,2},{0,2,3},{0,1,3}}; 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 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
 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 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

The Stanley Reisner ring of a simplicial 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}}; i9 : F={{0,1,2},{0,2,3},{0,1,3}}; i10 : R=QQ[x,y]; i11 : CF = courantFunctions(V,F,Homogenize=>false,BaseRing=>R); 3 3 o11 : Matrix R <-- R 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 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 i15 : V'={{0,1},{-1,-1},{1,-1},{1,10},{-2,-2},{2,-2}};--asymmetric triangular prism i16 : F={{0,1,2},{0,1,3,4},{0,2,3,5},{1,2,4,5}}; 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 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 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

• stanleyReisnerPresentation -- creates a ring map whose image is the sub-ring of $C^0(\Delta)$ generated by $C^r(\Delta)$. If $\Delta$ is simplicial, $C^0(\Delta)$ is the Stanley Reisner ring of $\Delta$.