Given a polynedron P we can associate a Toric variety Tv_P with coordinate ring A. Precisely A can be computed as 'A = teToricRing (newToricEmbedding {f0,...,f3})'. Assume P has p+1 lattice points, this is length latticePoints P = p+1, and denote with T_0 ... T_p the lattice points of P. Consider the ring QQ[T_0..T_p]. Let J be the toric ideal of P, then teToricRing gives coordinate ring QQ[T_0..T_p]/J
The polynomials f0,...,f3 induces a map homogeneous g=(g_0:...:g_3):Tv_P --> P^3
teToricRationalMap ({f0,...,f3},P) gives the map g=(g_0:...:g_3):Tv_P --> P^3 induced by the polynomials f0,...,f3
teToricRationalMap ({f0,...,f3}) gives the map g=(g_0:...:g_3):Tv_N --> P^3 induced by the polynomials f0,...,f3, where N = polynomialsToPolytope({f0,...,f3}).
teToricRationalMap ({f0,...,f3}) == teToricRationalMap ({f0,...,f3}, polynomialsToPolytope({f0,...,f3})).
i1 : needsPackage "Polyhedra" o1 = Polyhedra o1 : Package |
i2 : S = QQ[s,t]; |
i3 : f0 = s^2+s^3*t; |
i4 : f1 = s^3*t^6+1; |
i5 : f2 = s*t^2+2*s^3*t^5; |
i6 : f3 = s^2+s^3*t^6; |
i7 : l = {f0,f1,f2,f3};
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i8 : P = convexHull(matrix{{0,0,1},{0,1,0}});
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i9 : teToricRationalMap (l,P)
o9 = | T_0^7T_2^2+T_0^5T_1T_2^3 T_0^9+T_1^6T_2^3 T_0^6T_1^2T_2+2T_0T_1^5T_2^3
------------------------------------------------------------------------
T_0^7T_2^2+T_1^6T_2^3 |
1 4
o9 : Matrix (QQ[T , T , T ]) <--- (QQ[T , T , T ])
0 1 2 0 1 2
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i10 : S = QQ[s,t]; |
i11 : f0 = s^2+s^3*t; |
i12 : f1 = s^3*t^6+1; |
i13 : f2 = s*t^2+2*s^3*t^5; |
i14 : f3 = s^2+s^3*t^6; |
i15 : l = {f0,f1,f2,f3};
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i16 : G = teToricRationalMap(l, polynomialsToPolytope l);
/ QQ[T , T , T , T , T , T , T , T , T , T , T , T , T , T , T ] \ / QQ[T , T , T , T , T , T , T , T , T , T , T , T , T , T , T ] \
| 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 |1 | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 |4
o16 : Matrix |-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| <--- |-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
| 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 | | 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 |
|(T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T - T T , T T - T T , T T - T T , T - T T , T T - T T , T - T T , T T T - T T , T T T - T T T , T T T - T T , T T T - T T T , T T T - T T , T T T - T T T , T T T - T T , T T T - T T T , T T - T T , T T - T T T )| |(T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T T - T T , T - T T , T T - T T , T T - T T , T - T T , T T - T T , T - T T , T T T - T T , T T T - T T T , T T T - T T , T T T - T T T , T T T - T T , T T T - T T T , T T T - T T , T T T - T T T , T T - T T , T T - T T T )|
\ 13 12 14 12 13 11 14 11 13 10 14 10 13 9 14 8 13 7 14 7 13 6 14 6 13 5 14 5 13 4 14 3 13 2 14 2 13 1 14 12 10 14 11 12 9 14 10 12 9 13 8 12 6 14 7 12 5 14 6 12 4 14 5 12 4 13 3 12 1 14 2 12 1 13 11 9 13 10 11 9 12 8 11 5 14 7 11 4 14 6 11 4 13 5 11 4 12 3 11 1 13 2 11 1 12 10 9 11 8 10 4 14 7 10 4 13 6 10 4 12 5 10 4 11 3 10 1 12 2 10 1 11 8 9 4 13 7 9 4 12 6 9 4 11 5 9 4 10 3 9 1 11 2 9 1 10 8 3 14 7 8 2 14 6 8 1 14 5 8 1 13 4 8 1 12 3 8 0 14 2 8 0 13 1 8 0 12 7 1 14 6 7 1 13 5 7 1 12 4 7 1 11 3 7 0 13 2 7 0 12 1 7 0 11 6 1 12 5 6 1 11 4 6 1 10 3 6 0 12 2 6 0 11 1 6 0 10 5 1 10 4 5 1 9 3 5 0 11 2 5 0 10 1 5 0 9 3 4 0 10 2 4 0 9 3 0 8 2 3 0 7 1 3 0 6 2 0 6 1 2 0 5 1 0 4 1 9 13 4 14 0 9 13 1 4 14 1 9 12 4 13 0 9 12 1 4 13 1 9 11 4 12 0 9 11 1 4 12 1 9 10 4 11 0 9 10 1 4 11 1 9 4 10 0 9 1 4 10 / \ 13 12 14 12 13 11 14 11 13 10 14 10 13 9 14 8 13 7 14 7 13 6 14 6 13 5 14 5 13 4 14 3 13 2 14 2 13 1 14 12 10 14 11 12 9 14 10 12 9 13 8 12 6 14 7 12 5 14 6 12 4 14 5 12 4 13 3 12 1 14 2 12 1 13 11 9 13 10 11 9 12 8 11 5 14 7 11 4 14 6 11 4 13 5 11 4 12 3 11 1 13 2 11 1 12 10 9 11 8 10 4 14 7 10 4 13 6 10 4 12 5 10 4 11 3 10 1 12 2 10 1 11 8 9 4 13 7 9 4 12 6 9 4 11 5 9 4 10 3 9 1 11 2 9 1 10 8 3 14 7 8 2 14 6 8 1 14 5 8 1 13 4 8 1 12 3 8 0 14 2 8 0 13 1 8 0 12 7 1 14 6 7 1 13 5 7 1 12 4 7 1 11 3 7 0 13 2 7 0 12 1 7 0 11 6 1 12 5 6 1 11 4 6 1 10 3 6 0 12 2 6 0 11 1 6 0 10 5 1 10 4 5 1 9 3 5 0 11 2 5 0 10 1 5 0 9 3 4 0 10 2 4 0 9 3 0 8 2 3 0 7 1 3 0 6 2 0 6 1 2 0 5 1 0 4 1 9 13 4 14 0 9 13 1 4 14 1 9 12 4 13 0 9 12 1 4 13 1 9 11 4 12 0 9 11 1 4 12 1 9 10 4 11 0 9 10 1 4 11 1 9 4 10 0 9 1 4 10 /
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