next | previous | forward | backward | up | top | index | toc | Macaulay2 web site
MatrixRepToric :: teToricRationalMap

teToricRationalMap -- Computes the rational map G defined by {f0,f1,f2,f3} over the toric coordinate ring associated to a polytope P

Synopsis

Description

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};
i8 : P = convexHull(matrix{{0,0,1},{0,1,0}});
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
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}; 
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 /

See also

Ways to use teToricRationalMap :