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Cremona :: approximateInverseMap

approximateInverseMap -- random map related to the inverse of a birational map

Synopsis

Description

The algorithm is to try to construct the ideal of the base locus of the inverse by looking for the images via phi of random linear sections of the source variety. Generally, one can speed up the process by passing through the option CodimBsInv a good lower bound for the codimension of this base locus.

i1 : P8 = ZZ/97[t_0..t_8];
i2 : phi = inverseMap rationalMap(trim(minors(2,genericMatrix(P8,3,3))+random(2,P8)),Dominant=>true)

o2 = -- rational map --
                                ZZ
     source: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x ]) defined by
                                97  0   1   2   3   4   5   6   7   8   9
             {
               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                         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 2          2      2 2         2          2          2      2 2
              x x  + 14x x x  + 26x x  - 2x x x x  - 14x x x x  + 11x x x x  + 19x x x  + x x  - 11x x x  + 38x x  - 14x x x x  - 11x x x  + 45x x x x  - 19x x x x  + 14x x x  + 11x x x x  - 19x x x x  + 21x x x x  + 26x x  + 19x x x  + 38x x  - 30x x x  - 11x x x x  - 11x x x  + 30x x x x  + 22x x x x  + 11x x x x  - 22x x x  + 10x x x x  + 11x x x x  - 42x x x x  - 10x x x x  + 42x x x  - 38x x  - 37x x x  + 7x x  + 28x x x  - 38x x  - 3x x x  - 29x x x  - 47x x x  + 36x x  + 30x x x x  - 22x x x  - 20x x x x  - 41x x x  - 30x x x  + 22x x x x  + 20x x x x  - 4x x x  + 26x x x x  + 41x x x x  - 28x x x x  + 4x x x x  - 26x x x x  + 12x x x x  + 28x x x  - 12x x x  - 21x x x x  - 28x x x x  + 37x x x x  + 21x x x x  - 11x x x  - 28x x x x  - 21x x x x  + 3x x x x  + 47x x x x  + 48x x x x  + 2x x x x  - 45x x x x  - 33x x x  - 38x x  + 28x x x  - 38x x  - 21x x x  + 48x x  - 48x x x  + 45x x x  - 47x x x  - 48x x  + 11x x x x  - 10x x x  + 20x x x  + 11x x x x  + 42x x x  + 41x x x x  - 11x x x  + 10x x x x  - 20x x x x  - 26x x x x  + 28x x x x  + 4x x x x  + 26x x x  - 12x x x  - 11x x x  - 42x x x x  - 41x x x x  - 4x x x  - 28x x x x  + 12x x x x  + 37x x x x  + 3x x x x  - 21x x x  - 14x x x x  + 29x x x x  + 11x x x x  + 47x x x x  - 48x x x x  - 2x x x x  + 45x x x  + 29x x x x  + 25x x x x  + 33x x x x  - 37x x x  - 3x x x x  - 47x x x  + 21x x x x  + 11x x x x  + x x x x  + 48x x x x  - 45x x x x  + 47x x x x  - 2x x x x  + 33x x x x  + 47x x x x  - x x x x  + 7x x  - 29x x x  + 36x x  - 11x x x  + 48x x  + 2x x x  - 33x x x  - 47x x x  - 48x x  + 48x x x x  - 48x x x x  - 48x x x x  + 48x x x x  + 48x x x x  - 48x x x x
               1 2      1 2 3      1 3     0 1 2 4      0 1 3 4      1 2 3 4      1 3 4    0 4      0 3 4      3 4      0 1 2 5      1 2 5      0 1 3 5      1 2 3 5      0 4 5      0 2 4 5      0 3 4 5      2 3 4 5      0 5      0 2 5      2 5      1 2 6      1 2 3 6      1 3 6      0 2 4 6      1 2 4 6      0 3 4 6      0 4 6      1 2 5 6      0 3 5 6      1 3 5 6      0 4 5 6      0 5 6      2 6      2 3 6     3 6      2 4 6      4 6     2 5 6      3 5 6      4 5 6      5 6      0 1 2 7      1 2 7      1 2 3 7      1 3 7      0 4 7      0 1 4 7      0 3 4 7     3 4 7      1 2 5 7      0 3 5 7      1 3 5 7     2 3 5 7      0 4 5 7      3 4 5 7      0 5 7      2 5 7      0 2 6 7      1 2 6 7      0 3 6 7      2 3 6 7      3 6 7      0 4 6 7      1 4 6 7     0 5 6 7      1 5 6 7      2 5 6 7     3 5 6 7      4 5 6 7      5 6 7      0 7      0 1 7      1 7      0 3 7      3 7      0 5 7      1 5 7      3 5 7      5 7      0 1 2 8      1 2 8      1 2 8      0 1 3 8      1 3 8      1 2 3 8      0 4 8      0 1 4 8      0 2 4 8      1 2 4 8      1 3 4 8     2 3 4 8      0 4 8      3 4 8      0 5 8      0 1 5 8      0 2 5 8     2 5 8      0 4 5 8      2 4 5 8      0 2 6 8     1 2 6 8      2 6 8      0 3 6 8      1 3 6 8      2 3 6 8      1 4 6 8      2 4 6 8     3 4 6 8      4 6 8      0 5 6 8      1 5 6 8      4 5 6 8      0 7 8     0 1 7 8      1 7 8      0 2 7 8      0 3 7 8    2 3 7 8      0 4 7 8      1 4 7 8      3 4 7 8     0 5 7 8      1 5 7 8      2 5 7 8    4 5 7 8     0 8      0 1 8      1 8      0 2 8      2 8     0 4 8      1 4 8      2 4 8      4 8      3 4 6 9      2 5 6 9      1 3 7 9      0 5 7 9      1 2 8 9      0 4 8 9
             }
                  ZZ
     target: Proj(--[t , t , t , t , t , t , t , t , t ])
                  97  0   1   2   3   4   5   6   7   8
     defining forms: {
                      x x  - x x ,
                       5 7    4 8
                      
                      x x  - x x ,
                       5 6    1 8
                      
                      x x  - x x ,
                       4 6    1 7
                      
                      x x  - x x ,
                       3 7    2 8
                      
                      x x  - x x ,
                       3 6    0 8
                      
                      x x  - x x ,
                       2 6    0 7
                      
                      x x  - x x ,
                       3 4    2 5
                      
                      x x  - x x ,
                       1 3    0 5
                      
                      x x  - x x
                       1 2    0 4
                     }

o2 : RationalMap (quadratic rational map from hypersurface in PP^9 to PP^8)
i3 : time psi = approximateInverseMap phi
-- approximateInverseMap: step 1 of 10
-- approximateInverseMap: step 2 of 10
-- approximateInverseMap: step 3 of 10
-- approximateInverseMap: step 4 of 10
-- approximateInverseMap: step 5 of 10
-- approximateInverseMap: step 6 of 10
-- approximateInverseMap: step 7 of 10
-- approximateInverseMap: step 8 of 10
-- approximateInverseMap: step 9 of 10
-- approximateInverseMap: step 10 of 10
     -- used 1.06026 seconds

o3 = -- rational map --
                  ZZ
     source: Proj(--[t , t , t , t , t , t , t , t , t ])
                  97  0   1   2   3   4   5   6   7   8
                                ZZ
     target: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x ]) defined by
                                97  0   1   2   3   4   5   6   7   8   9
             {
               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                         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 2          2      2 2         2          2          2      2 2
              x x  + 14x x x  + 26x x  - 2x x x x  - 14x x x x  + 11x x x x  + 19x x x  + x x  - 11x x x  + 38x x  - 14x x x x  - 11x x x  + 45x x x x  - 19x x x x  + 14x x x  + 11x x x x  - 19x x x x  + 21x x x x  + 26x x  + 19x x x  + 38x x  - 30x x x  - 11x x x x  - 11x x x  + 30x x x x  + 22x x x x  + 11x x x x  - 22x x x  + 10x x x x  + 11x x x x  - 42x x x x  - 10x x x x  + 42x x x  - 38x x  - 37x x x  + 7x x  + 28x x x  - 38x x  - 3x x x  - 29x x x  - 47x x x  + 36x x  + 30x x x x  - 22x x x  - 20x x x x  - 41x x x  - 30x x x  + 22x x x x  + 20x x x x  - 4x x x  + 26x x x x  + 41x x x x  - 28x x x x  + 4x x x x  - 26x x x x  + 12x x x x  + 28x x x  - 12x x x  - 21x x x x  - 28x x x x  + 37x x x x  + 21x x x x  - 11x x x  - 28x x x x  - 21x x x x  + 3x x x x  + 47x x x x  + 48x x x x  + 2x x x x  - 45x x x x  - 33x x x  - 38x x  + 28x x x  - 38x x  - 21x x x  + 48x x  - 48x x x  + 45x x x  - 47x x x  - 48x x  + 11x x x x  - 10x x x  + 20x x x  + 11x x x x  + 42x x x  + 41x x x x  - 11x x x  + 10x x x x  - 20x x x x  - 26x x x x  + 28x x x x  + 4x x x x  + 26x x x  - 12x x x  - 11x x x  - 42x x x x  - 41x x x x  - 4x x x  - 28x x x x  + 12x x x x  + 37x x x x  + 3x x x x  - 21x x x  - 14x x x x  + 29x x x x  + 11x x x x  + 47x x x x  - 48x x x x  - 2x x x x  + 45x x x  + 29x x x x  + 25x x x x  + 33x x x x  - 37x x x  - 3x x x x  - 47x x x  + 21x x x x  + 11x x x x  + x x x x  + 48x x x x  - 45x x x x  + 47x x x x  - 2x x x x  + 33x x x x  + 47x x x x  - x x x x  + 7x x  - 29x x x  + 36x x  - 11x x x  + 48x x  + 2x x x  - 33x x x  - 47x x x  - 48x x  + 48x x x x  - 48x x x x  - 48x x x x  + 48x x x x  + 48x x x x  - 48x x x x
               1 2      1 2 3      1 3     0 1 2 4      0 1 3 4      1 2 3 4      1 3 4    0 4      0 3 4      3 4      0 1 2 5      1 2 5      0 1 3 5      1 2 3 5      0 4 5      0 2 4 5      0 3 4 5      2 3 4 5      0 5      0 2 5      2 5      1 2 6      1 2 3 6      1 3 6      0 2 4 6      1 2 4 6      0 3 4 6      0 4 6      1 2 5 6      0 3 5 6      1 3 5 6      0 4 5 6      0 5 6      2 6      2 3 6     3 6      2 4 6      4 6     2 5 6      3 5 6      4 5 6      5 6      0 1 2 7      1 2 7      1 2 3 7      1 3 7      0 4 7      0 1 4 7      0 3 4 7     3 4 7      1 2 5 7      0 3 5 7      1 3 5 7     2 3 5 7      0 4 5 7      3 4 5 7      0 5 7      2 5 7      0 2 6 7      1 2 6 7      0 3 6 7      2 3 6 7      3 6 7      0 4 6 7      1 4 6 7     0 5 6 7      1 5 6 7      2 5 6 7     3 5 6 7      4 5 6 7      5 6 7      0 7      0 1 7      1 7      0 3 7      3 7      0 5 7      1 5 7      3 5 7      5 7      0 1 2 8      1 2 8      1 2 8      0 1 3 8      1 3 8      1 2 3 8      0 4 8      0 1 4 8      0 2 4 8      1 2 4 8      1 3 4 8     2 3 4 8      0 4 8      3 4 8      0 5 8      0 1 5 8      0 2 5 8     2 5 8      0 4 5 8      2 4 5 8      0 2 6 8     1 2 6 8      2 6 8      0 3 6 8      1 3 6 8      2 3 6 8      1 4 6 8      2 4 6 8     3 4 6 8      4 6 8      0 5 6 8      1 5 6 8      4 5 6 8      0 7 8     0 1 7 8      1 7 8      0 2 7 8      0 3 7 8    2 3 7 8      0 4 7 8      1 4 7 8      3 4 7 8     0 5 7 8      1 5 7 8      2 5 7 8    4 5 7 8     0 8      0 1 8      1 8      0 2 8      2 8     0 4 8      1 4 8      2 4 8      4 8      3 4 6 9      2 5 6 9      1 3 7 9      0 5 7 9      1 2 8 9      0 4 8 9
             }
     defining forms: {
                      t t  - t t ,
                       5 7    4 8
                      
                      t t  - t t ,
                       2 7    1 8
                      
                      t t  - t t ,
                       5 6    3 8
                      
                      t t  - t t ,
                       4 6    3 7
                      
                      t t  - t t ,
                       2 6    0 8
                      
                      t t  - t t ,
                       1 6    0 7
                      
                      t t  - t t ,
                       2 4    1 5
                      
                      t t  - t t ,
                       2 3    0 5
                      
                      t t  - t t ,
                       1 3    0 4
                      
                       2               2                      2            2                                2                                                2                       2                                                   2                                                                             2
                      t  + 31t t  - 25t  + 7t t  + 3t t  + 21t  + 3t t  - t  + 4t t  + 39t t  - 22t t  + 14t  - t t  - 6t t  - 41t t  + 42t t  + 23t t  + 21t  + 24t t  - 8t t  - 21t  + 41t t  + 13t t  + 15t t  - 22t t  + 38t t  - 45t  - 45t t  + 20t t  + 44t t  - 40t t  - 22t t  + 37t t  + 22t t  + 28t t  + 2t
                       0      0 1      1     0 2     1 2      2     0 3    3     0 4      1 4      3 4      4    0 5     1 5      2 5      3 5      4 5      5      0 6     3 6      6      0 7      1 7      3 7      4 7      6 7      7      0 8      1 8      2 8      3 8      4 8      5 8      6 8      7 8     8
                     }

o3 : RationalMap (quadratic rational map from PP^8 to hypersurface in PP^9)
i4 : assert(phi * psi == 1 and psi * phi == 1)
i5 : time psi' = approximateInverseMap(phi,CodimBsInv=>5);
-- approximateInverseMap: step 1 of 3
-- approximateInverseMap: step 2 of 3
-- approximateInverseMap: step 3 of 3
     -- used 0.563491 seconds

o5 : RationalMap (quadratic rational map from PP^8 to hypersurface in PP^9)
i6 : assert(psi == psi')

A more complicated example is the following (here inverseMap takes a lot of time!).

i7 : phi = rationalMap map(P8,ZZ/97[x_0..x_11]/ideal(x_1*x_3-8*x_2*x_3+25*x_3^2-25*x_2*x_4-22*x_3*x_4+x_0*x_5+13*x_2*x_5+41*x_3*x_5-x_0*x_6+12*x_2*x_6+25*x_1*x_7+25*x_3*x_7+23*x_5*x_7-3*x_6*x_7+2*x_0*x_8+11*x_1*x_8-37*x_3*x_8-23*x_4*x_8-33*x_6*x_8+8*x_0*x_9+10*x_1*x_9-25*x_2*x_9-9*x_3*x_9+3*x_4*x_9+24*x_5*x_9-27*x_6*x_9-5*x_0*x_10+28*x_1*x_10+37*x_2*x_10+9*x_4*x_10+27*x_6*x_10-25*x_0*x_11+9*x_2*x_11+27*x_4*x_11-27*x_5*x_11,x_2^2+17*x_2*x_3-14*x_3^2-13*x_2*x_4+34*x_3*x_4+44*x_0*x_5-30*x_2*x_5+27*x_3*x_5+31*x_2*x_6-36*x_3*x_6-x_0*x_7+13*x_1*x_7+8*x_3*x_7+9*x_5*x_7+46*x_6*x_7+41*x_0*x_8-7*x_1*x_8-34*x_3*x_8-9*x_4*x_8-46*x_6*x_8-17*x_0*x_9+32*x_1*x_9-8*x_2*x_9-35*x_3*x_9-46*x_4*x_9+26*x_5*x_9+17*x_6*x_9+15*x_0*x_10+35*x_1*x_10+34*x_2*x_10+20*x_4*x_10+14*x_0*x_11+36*x_1*x_11+35*x_2*x_11-17*x_4*x_11,x_1*x_2-40*x_2*x_3+28*x_3^2-x_0*x_4+5*x_2*x_4-16*x_3*x_4+5*x_0*x_5-36*x_2*x_5+37*x_3*x_5+48*x_2*x_6-5*x_1*x_7-5*x_3*x_7+x_5*x_7+20*x_6*x_7+10*x_0*x_8+34*x_1*x_8+41*x_3*x_8-x_4*x_8+x_6*x_8+40*x_0*x_9-32*x_1*x_9+5*x_2*x_9-11*x_3*x_9-20*x_4*x_9+45*x_5*x_9-14*x_6*x_9-25*x_0*x_10+45*x_1*x_10-41*x_2*x_10-46*x_4*x_10+8*x_6*x_10-28*x_0*x_11+11*x_2*x_11+14*x_4*x_11-8*x_5*x_11),{t_4^2+t_0*t_5+t_1*t_5+35*t_2*t_5+10*t_3*t_5+25*t_4*t_5-5*t_5^2-14*t_0*t_6-14*t_1*t_6-5*t_2*t_6-13*t_4*t_6+37*t_5*t_6+22*t_6^2-31*t_3*t_7+26*t_4*t_7+12*t_5*t_7-45*t_6*t_7-46*t_3*t_8+37*t_4*t_8+28*t_5*t_8+33*t_6*t_8,t_3*t_4+4*t_0*t_5+39*t_1*t_5-40*t_2*t_5+40*t_3*t_5+26*t_4*t_5-20*t_5^2+41*t_0*t_6+36*t_1*t_6-22*t_2*t_6+36*t_4*t_6-30*t_5*t_6-13*t_6^2-25*t_3*t_7+5*t_4*t_7-35*t_5*t_7+10*t_6*t_7+11*t_3*t_8+46*t_4*t_8+29*t_5*t_8+28*t_6*t_8,t_2*t_4-5*t_0*t_5-40*t_1*t_5+12*t_2*t_5+47*t_3*t_5+37*t_4*t_5+25*t_5^2-27*t_0*t_6-22*t_1*t_6+27*t_2*t_6-23*t_4*t_6+5*t_5*t_6-13*t_6^2-39*t_3*t_7-29*t_4*t_7+9*t_5*t_7+39*t_6*t_7+36*t_3*t_8+13*t_4*t_8+26*t_5*t_8+37*t_6*t_8,t_0*t_4-t_0*t_5-8*t_1*t_5-35*t_2*t_5-10*t_3*t_5-33*t_4*t_5+5*t_5^2+15*t_0*t_6+15*t_1*t_6+5*t_2*t_6+15*t_4*t_6-38*t_5*t_6-22*t_6^2+31*t_3*t_7-25*t_4*t_7-19*t_5*t_7+47*t_6*t_7+46*t_3*t_8-36*t_4*t_8-35*t_5*t_8-31*t_6*t_8,t_2*t_3-t_0*t_5-t_1*t_5-35*t_2*t_5-10*t_3*t_5-33*t_4*t_5+5*t_5^2+14*t_0*t_6+14*t_1*t_6+5*t_2*t_6+14*t_4*t_6-31*t_5*t_6-24*t_6^2+32*t_3*t_7-25*t_4*t_7-19*t_5*t_7+47*t_6*t_7+46*t_3*t_8-36*t_4*t_8-35*t_5*t_8-31*t_6*t_8,t_1*t_3-7*t_1*t_5+t_1*t_6+t_4*t_6-7*t_5*t_6+2*t_6^2-t_3*t_7,t_0*t_3-46*t_0*t_5-39*t_1*t_5-43*t_2*t_5-41*t_3*t_5-26*t_4*t_5-28*t_5^2-35*t_0*t_6-36*t_1*t_6+20*t_2*t_6-36*t_4*t_6+9*t_5*t_6+15*t_6^2+26*t_3*t_7-5*t_4*t_7+35*t_5*t_7-10*t_6*t_7-10*t_3*t_8-46*t_4*t_8+47*t_5*t_8-25*t_6*t_8,t_2^2-46*t_1*t_4-33*t_0*t_5-45*t_1*t_5-39*t_2*t_5-39*t_3*t_5-46*t_4*t_5-29*t_5^2-48*t_0*t_6-38*t_1*t_6-30*t_2*t_6+19*t_4*t_6-44*t_5*t_6-47*t_6^2-36*t_0*t_7-46*t_1*t_7+t_2*t_7-44*t_3*t_7+48*t_4*t_7-14*t_5*t_7+4*t_6*t_7-36*t_0*t_8-46*t_1*t_8+47*t_2*t_8-34*t_3*t_8-24*t_4*t_8-12*t_5*t_8-47*t_6*t_8+47*t_7*t_8,t_1*t_2+6*t_1*t_5+5*t_0*t_6-2*t_1*t_6-t_4*t_6-t_5*t_6+5*t_0*t_7+t_1*t_7-2*t_2*t_7-7*t_5*t_7+2*t_6*t_7-2*t_1*t_8+3*t_7*t_8,t_0*t_2+t_1*t_4+5*t_0*t_5+32*t_1*t_5-20*t_2*t_5-47*t_3*t_5-37*t_4*t_5-25*t_5^2+19*t_0*t_6+22*t_1*t_6-25*t_2*t_6+25*t_4*t_6-5*t_5*t_6+13*t_6^2+5*t_0*t_7+t_1*t_7+39*t_3*t_7+28*t_4*t_7-9*t_5*t_7-39*t_6*t_7+4*t_0*t_8+t_1*t_8-36*t_3*t_8-14*t_4*t_8-26*t_5*t_8-37*t_6*t_8,t_0*t_1-39*t_1*t_4+40*t_1*t_5-37*t_0*t_6-39*t_1*t_6+19*t_4*t_6-39*t_5*t_6-38*t_0*t_7+39*t_1*t_7+19*t_2*t_7+18*t_5*t_7-19*t_6*t_7+19*t_1*t_8+20*t_7*t_8,t_0^2+12*t_1*t_4+20*t_0*t_5+27*t_1*t_5-8*t_2*t_5+37*t_3*t_5+28*t_4*t_5+30*t_5^2-46*t_0*t_6+24*t_1*t_6-40*t_2*t_6+25*t_4*t_6+16*t_5*t_6-35*t_6^2+29*t_0*t_7+12*t_1*t_7-35*t_2*t_7-8*t_3*t_7-18*t_4*t_7+42*t_5*t_7-12*t_6*t_7-6*t_0*t_8+12*t_1*t_8-15*t_3*t_8+9*t_4*t_8+20*t_5*t_8-30*t_6*t_8+4*t_7*t_8})

o7 = -- rational map --
                  ZZ
     source: Proj(--[t , t , t , t , t , t , t , t , t ])
                  97  0   1   2   3   4   5   6   7   8
                                ZZ
     target: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x , x  , x  ]) defined by
                                97  0   1   2   3   4   5   6   7   8   9   10   11
             {
                                2
              x x  - 8x x  + 25x  - 25x x  - 22x x  + x x  + 13x x  + 41x x  - x x  + 12x x  + 25x x  + 25x x  + 23x x  - 3x x  + 2x x  + 11x x  - 37x x  - 23x x  - 33x x  + 8x x  + 10x x  - 25x x  - 9x x  + 3x x  + 24x x  - 27x x  - 5x x   + 28x x   + 37x x   + 9x x   + 27x x   - 25x x   + 9x x   + 27x x   - 27x x  ,
               1 3     2 3      3      2 4      3 4    0 5      2 5      3 5    0 6      2 6      1 7      3 7      5 7     6 7     0 8      1 8      3 8      4 8      6 8     0 9      1 9      2 9     3 9     4 9      5 9      6 9     0 10      1 10      2 10     4 10      6 10      0 11     2 11      4 11      5 11
              
               2               2
              x  + 17x x  - 14x  - 13x x  + 34x x  + 44x x  - 30x x  + 27x x  + 31x x  - 36x x  - x x  + 13x x  + 8x x  + 9x x  + 46x x  + 41x x  - 7x x  - 34x x  - 9x x  - 46x x  - 17x x  + 32x x  - 8x x  - 35x x  - 46x x  + 26x x  + 17x x  + 15x x   + 35x x   + 34x x   + 20x x   + 14x x   + 36x x   + 35x x   - 17x x  ,
               2      2 3      3      2 4      3 4      0 5      2 5      3 5      2 6      3 6    0 7      1 7     3 7     5 7      6 7      0 8     1 8      3 8     4 8      6 8      0 9      1 9     2 9      3 9      4 9      5 9      6 9      0 10      1 10      2 10      4 10      0 11      1 11      2 11      4 11
              
                                 2
              x x  - 40x x  + 28x  - x x  + 5x x  - 16x x  + 5x x  - 36x x  + 37x x  + 48x x  - 5x x  - 5x x  + x x  + 20x x  + 10x x  + 34x x  + 41x x  - x x  + x x  + 40x x  - 32x x  + 5x x  - 11x x  - 20x x  + 45x x  - 14x x  - 25x x   + 45x x   - 41x x   - 46x x   + 8x x   - 28x x   + 11x x   + 14x x   - 8x x
               1 2      2 3      3    0 4     2 4      3 4     0 5      2 5      3 5      2 6     1 7     3 7    5 7      6 7      0 8      1 8      3 8    4 8    6 8      0 9      1 9     2 9      3 9      4 9      5 9      6 9      0 10      1 10      2 10      4 10     6 10      0 11      2 11      4 11     5 11
             }
     defining forms: {
                       2                                              2                                                  2
                      t  + t t  + t t  + 35t t  + 10t t  + 25t t  - 5t  - 14t t  - 14t t  - 5t t  - 13t t  + 37t t  + 22t  - 31t t  + 26t t  + 12t t  - 45t t  - 46t t  + 37t t  + 28t t  + 33t t ,
                       4    0 5    1 5      2 5      3 5      4 5     5      0 6      1 6     2 6      4 6      5 6      6      3 7      4 7      5 7      6 7      3 8      4 8      5 8      6 8
                      
                                                                            2                                                   2
                      t t  + 4t t  + 39t t  - 40t t  + 40t t  + 26t t  - 20t  + 41t t  + 36t t  - 22t t  + 36t t  - 30t t  - 13t  - 25t t  + 5t t  - 35t t  + 10t t  + 11t t  + 46t t  + 29t t  + 28t t ,
                       3 4     0 5      1 5      2 5      3 5      4 5      5      0 6      1 6      2 6      4 6      5 6      6      3 7     4 7      5 7      6 7      3 8      4 8      5 8      6 8
                      
                                                                            2                                                  2
                      t t  - 5t t  - 40t t  + 12t t  + 47t t  + 37t t  + 25t  - 27t t  - 22t t  + 27t t  - 23t t  + 5t t  - 13t  - 39t t  - 29t t  + 9t t  + 39t t  + 36t t  + 13t t  + 26t t  + 37t t ,
                       2 4     0 5      1 5      2 5      3 5      4 5      5      0 6      1 6      2 6      4 6     5 6      6      3 7      4 7     5 7      6 7      3 8      4 8      5 8      6 8
                      
                                                                         2                                                  2
                      t t  - t t  - 8t t  - 35t t  - 10t t  - 33t t  + 5t  + 15t t  + 15t t  + 5t t  + 15t t  - 38t t  - 22t  + 31t t  - 25t t  - 19t t  + 47t t  + 46t t  - 36t t  - 35t t  - 31t t ,
                       0 4    0 5     1 5      2 5      3 5      4 5     5      0 6      1 6     2 6      4 6      5 6      6      3 7      4 7      5 7      6 7      3 8      4 8      5 8      6 8
                      
                                                                        2                                                  2
                      t t  - t t  - t t  - 35t t  - 10t t  - 33t t  + 5t  + 14t t  + 14t t  + 5t t  + 14t t  - 31t t  - 24t  + 32t t  - 25t t  - 19t t  + 47t t  + 46t t  - 36t t  - 35t t  - 31t t ,
                       2 3    0 5    1 5      2 5      3 5      4 5     5      0 6      1 6     2 6      4 6      5 6      6      3 7      4 7      5 7      6 7      3 8      4 8      5 8      6 8
                      
                                                             2
                      t t  - 7t t  + t t  + t t  - 7t t  + 2t  - t t ,
                       1 3     1 5    1 6    4 6     5 6     6    3 7
                      
                                                                             2                                                  2
                      t t  - 46t t  - 39t t  - 43t t  - 41t t  - 26t t  - 28t  - 35t t  - 36t t  + 20t t  - 36t t  + 9t t  + 15t  + 26t t  - 5t t  + 35t t  - 10t t  - 10t t  - 46t t  + 47t t  - 25t t ,
                       0 3      0 5      1 5      2 5      3 5      4 5      5      0 6      1 6      2 6      4 6     5 6      6      3 7     4 7      5 7      6 7      3 8      4 8      5 8      6 8
                      
                       2                                                            2                                                   2
                      t  - 46t t  - 33t t  - 45t t  - 39t t  - 39t t  - 46t t  - 29t  - 48t t  - 38t t  - 30t t  + 19t t  - 44t t  - 47t  - 36t t  - 46t t  + t t  - 44t t  + 48t t  - 14t t  + 4t t  - 36t t  - 46t t  + 47t t  - 34t t  - 24t t  - 12t t  - 47t t  + 47t t ,
                       2      1 4      0 5      1 5      2 5      3 5      4 5      5      0 6      1 6      2 6      4 6      5 6      6      0 7      1 7    2 7      3 7      4 7      5 7     6 7      0 8      1 8      2 8      3 8      4 8      5 8      6 8      7 8
                      
                      t t  + 6t t  + 5t t  - 2t t  - t t  - t t  + 5t t  + t t  - 2t t  - 7t t  + 2t t  - 2t t  + 3t t ,
                       1 2     1 5     0 6     1 6    4 6    5 6     0 7    1 7     2 7     5 7     6 7     1 8     7 8
                      
                                                                                   2                                                  2
                      t t  + t t  + 5t t  + 32t t  - 20t t  - 47t t  - 37t t  - 25t  + 19t t  + 22t t  - 25t t  + 25t t  - 5t t  + 13t  + 5t t  + t t  + 39t t  + 28t t  - 9t t  - 39t t  + 4t t  + t t  - 36t t  - 14t t  - 26t t  - 37t t ,
                       0 2    1 4     0 5      1 5      2 5      3 5      4 5      5      0 6      1 6      2 6      4 6     5 6      6     0 7    1 7      3 7      4 7     5 7      6 7     0 8    1 8      3 8      4 8      5 8      6 8
                      
                      t t  - 39t t  + 40t t  - 37t t  - 39t t  + 19t t  - 39t t  - 38t t  + 39t t  + 19t t  + 18t t  - 19t t  + 19t t  + 20t t ,
                       0 1      1 4      1 5      0 6      1 6      4 6      5 6      0 7      1 7      2 7      5 7      6 7      1 8      7 8
                      
                       2                                                           2                                                   2
                      t  + 12t t  + 20t t  + 27t t  - 8t t  + 37t t  + 28t t  + 30t  - 46t t  + 24t t  - 40t t  + 25t t  + 16t t  - 35t  + 29t t  + 12t t  - 35t t  - 8t t  - 18t t  + 42t t  - 12t t  - 6t t  + 12t t  - 15t t  + 9t t  + 20t t  - 30t t  + 4t t
                       0      1 4      0 5      1 5     2 5      3 5      4 5      5      0 6      1 6      2 6      4 6      5 6      6      0 7      1 7      2 7     3 7      4 7      5 7      6 7     0 8      1 8      3 8     4 8      5 8      6 8     7 8
                     }

o7 : RationalMap (quadratic rational map from PP^8 to 8-dimensional subvariety of PP^11)
i8 : -- without the option 'CodimBsInv=>4', it takes about triple time 
     time psi=approximateInverseMap(phi,CodimBsInv=>4)
-- approximateInverseMap: step 1 of 3
-- approximateInverseMap: step 2 of 3
-- approximateInverseMap: step 3 of 3
     -- used 2.71396 seconds

o8 = -- rational map --
                                ZZ
     source: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x , x  , x  ]) defined by
                                97  0   1   2   3   4   5   6   7   8   9   10   11
             {
                                2
              x x  - 8x x  + 25x  - 25x x  - 22x x  + x x  + 13x x  + 41x x  - x x  + 12x x  + 25x x  + 25x x  + 23x x  - 3x x  + 2x x  + 11x x  - 37x x  - 23x x  - 33x x  + 8x x  + 10x x  - 25x x  - 9x x  + 3x x  + 24x x  - 27x x  - 5x x   + 28x x   + 37x x   + 9x x   + 27x x   - 25x x   + 9x x   + 27x x   - 27x x  ,
               1 3     2 3      3      2 4      3 4    0 5      2 5      3 5    0 6      2 6      1 7      3 7      5 7     6 7     0 8      1 8      3 8      4 8      6 8     0 9      1 9      2 9     3 9     4 9      5 9      6 9     0 10      1 10      2 10     4 10      6 10      0 11     2 11      4 11      5 11
              
               2               2
              x  + 17x x  - 14x  - 13x x  + 34x x  + 44x x  - 30x x  + 27x x  + 31x x  - 36x x  - x x  + 13x x  + 8x x  + 9x x  + 46x x  + 41x x  - 7x x  - 34x x  - 9x x  - 46x x  - 17x x  + 32x x  - 8x x  - 35x x  - 46x x  + 26x x  + 17x x  + 15x x   + 35x x   + 34x x   + 20x x   + 14x x   + 36x x   + 35x x   - 17x x  ,
               2      2 3      3      2 4      3 4      0 5      2 5      3 5      2 6      3 6    0 7      1 7     3 7     5 7      6 7      0 8     1 8      3 8     4 8      6 8      0 9      1 9     2 9      3 9      4 9      5 9      6 9      0 10      1 10      2 10      4 10      0 11      1 11      2 11      4 11
              
                                 2
              x x  - 40x x  + 28x  - x x  + 5x x  - 16x x  + 5x x  - 36x x  + 37x x  + 48x x  - 5x x  - 5x x  + x x  + 20x x  + 10x x  + 34x x  + 41x x  - x x  + x x  + 40x x  - 32x x  + 5x x  - 11x x  - 20x x  + 45x x  - 14x x  - 25x x   + 45x x   - 41x x   - 46x x   + 8x x   - 28x x   + 11x x   + 14x x   - 8x x
               1 2      2 3      3    0 4     2 4      3 4     0 5      2 5      3 5      2 6     1 7     3 7    5 7      6 7      0 8      1 8      3 8    4 8    6 8      0 9      1 9     2 9      3 9      4 9      5 9      6 9      0 10      1 10      2 10      4 10     6 10      0 11      2 11      4 11     5 11
             }
                  ZZ
     target: Proj(--[t , t , t , t , t , t , t , t , t ])
                  97  0   1   2   3   4   5   6   7   8
     defining forms: {
                      x x  + 2x x  + x x  + 2x x  - 4x x  - 2x x  + x x  - 6x x  - 5x x   - 10x x   + 25x x   + 5x x   - 5x x  ,
                       3 5     5 7    1 8     2 8     3 8     4 8    5 8     5 9     1 10      2 10      3 10     4 10     5 10
                      
                      x x  + 6x x  + 2x x  + 5x x  - 10x x  - 6x x  - 46x x  - 44x x  - 22x x  + 36x x   - 25x x   + 14x x   + 15x x   + 36x x   + 36x x   - 36x x  ,
                       2 5     5 7     1 8     2 8      3 8     4 8      5 8      6 8      5 9      1 10      2 10      3 10      4 10      5 10      6 10      5 11
                      
                      x x  + 8x x  + x x  + 4x x  + 10x x  - 25x x  - 8x x  + 5x x  - 25x x  - 20x x   - 45x x   + 28x x   + 20x x   - 25x x  ,
                       0 5     5 7    0 8     1 8      2 8      3 8     4 8     5 8      5 9      1 10      2 10      3 10      4 10      5 10
                      
                      x x  - 4x x  + 46x x  - 38x x  - 43x x  - 39x x  - 41x x  + 19x x  + 40x x  + 43x x  - 39x x  - 37x x  + 3x x  + 38x x  - 2x x  + 39x x  - 11x x  - 28x x   + 21x x   - 5x x   - 10x x   + x x   - x x   - 46x x   + 2x x   + x x  ,
                       3 4     2 6      3 6      3 7      5 7      6 7      1 8      2 8      3 8      4 8      5 8      6 8     1 9      2 9     3 9      4 9      5 9      1 10      2 10     3 10      4 10    5 10    6 10      1 11     2 11    5 11
                      
                      x x  + 17x x  + 41x x  - x x  - 18x x  + 46x x  - 16x x  - 36x x  + 9x x  - 27x x  - 46x x  - 44x x  - 29x x  - 17x x  + 18x x  + 45x x  + 16x x  + 5x x  + 36x x  + 39x x   - 36x x   - 33x x   - 29x x   + 26x x   - 26x x   - 41x x   - 45x x   - 36x x   + 26x x  ,
                       2 4      2 6      3 6    1 7      3 7      5 7      6 7      1 8     2 8      3 8      4 8      5 8      6 8      1 9      2 9      3 9      4 9     5 9      6 9      1 10      2 10      3 10      4 10      5 10      6 10      1 11      2 11      4 11      5 11
                      
                       2
                      x  - 32x x  + 42x x  + 27x x  - 42x x  + 33x x  + 33x x  - 21x x  - 25x x  + 45x x  + 23x x  + 10x x  - 7x x  + 5x x  - 7x x  + 45x x  + 25x x  - 47x x  + 4x x  - 29x x  + 32x x  - 13x x  - 47x x  - 42x x  - 18x x   - 5x x   + 44x x   - 39x x   - 15x x   - 31x x   + 15x x   - 17x x   - 29x x   + 35x x   - 17x x   - 31x x  ,
                       3      0 4      0 6      2 6      3 6      0 7      2 7      3 7      5 7      6 7      0 8      1 8     2 8     3 8     4 8      5 8      6 8      0 9     1 9      2 9      3 9      4 9      5 9      6 9      0 10     1 10      2 10      3 10      4 10      5 10      6 10      0 11      1 11      2 11      3 11      5 11
                      
                      x x  + x x  - 16x x  - 10x x  + 2x x  + 2x x  + 34x x  + 22x x  + 38x x  + x x  + 30x x  - 21x x  - 34x x  - 21x x  + 38x x  + 46x x  - 6x x  + 12x x  - 48x x  + 17x x  - 39x x  - 44x x  - 5x x   - 15x x   - 13x x   - 20x x   - 45x x   + 4x x   - 4x x   + 10x x   + 8x x   + 4x x  ,
                       2 3    0 4      2 6      3 6     0 7     2 7      3 7      5 7      6 7    0 8      1 8      2 8      3 8      4 8      5 8      6 8     0 9      1 9      2 9      3 9      4 9      5 9     0 10      1 10      2 10      3 10      4 10     5 10     6 10      1 11     2 11     5 11
                      
                       2
                      x  + 36x x  + x x  + 7x x  - 4x x  - 21x x  + 30x x  - x x  - 5x x  - 29x x  + 2x x  - 29x x  - 27x x  - 5x x  + 4x x  + 17x x  - 44x x  + 18x x  - 42x x  + 39x x  + 9x x  - 13x x  + 48x x  - 32x x  - 37x x  - 4x x  - 38x x  + 10x x  + 13x x  + 12x x   + 35x x   + 13x x   + 31x x   + 30x x   - 45x x   - 15x x   + 37x x   + 44x x   + 7x x   + 9x x   - 25x x   - 45x x  ,
                       1      0 4    4 5     0 6     1 6      2 6      3 6    4 6     5 6      0 7     1 7      2 7      3 7     5 7     6 7      0 8      1 8      2 8      3 8      4 8     5 8      6 8      0 9      1 9      2 9     3 9      4 9      5 9      6 9      0 10      1 10      2 10      3 10      4 10      5 10      6 10      0 11      1 11     2 11     3 11      4 11      5 11
                      
                      x x  + 2x x  - 5x x  - 34x x  - 16x x  + 31x x  + 29x x  + 10x x  - 27x x  - 31x x  - 17x x  - 28x x  + 39x x  - 32x x  - 7x x  - 29x x  - 47x x  + 35x x  + 38x x  - 43x x  - 2x x  - 32x x  - 43x x  + 12x x  - 12x x   + 14x x   - 47x x   - 46x x   - 14x x   + 48x x   + 12x x   - 12x x   + 48x x   - x x   - 9x x   + 48x x
                       0 1     0 2     0 3      0 4      0 6      0 7      2 7      3 7      5 7      6 7      0 8      1 8      2 8      3 8     4 8      5 8      6 8      0 9      1 9      2 9     3 9      4 9      5 9      6 9      0 10      1 10      2 10      3 10      4 10      5 10      6 10      0 11      1 11    2 11     3 11      5 11
                     }

o8 : RationalMap (quadratic rational map from 8-dimensional subvariety of PP^11 to PP^8)
i9 : -- but...
     phi * psi == 1

o9 = false
i10 : -- in this case we can remedy enabling the option MathMode
      time psi = approximateInverseMap(phi,CodimBsInv=>4,MathMode=>true)
-- approximateInverseMap: step 1 of 3
-- approximateInverseMap: step 2 of 3
-- approximateInverseMap: step 3 of 3
MathMode: output certified!
     -- used 3.80301 seconds

o10 = -- rational map --
                                 ZZ
      source: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x , x  , x  ]) defined by
                                 97  0   1   2   3   4   5   6   7   8   9   10   11
              {
                                 2
               x x  - 8x x  + 25x  - 25x x  - 22x x  + x x  + 13x x  + 41x x  - x x  + 12x x  + 25x x  + 25x x  + 23x x  - 3x x  + 2x x  + 11x x  - 37x x  - 23x x  - 33x x  + 8x x  + 10x x  - 25x x  - 9x x  + 3x x  + 24x x  - 27x x  - 5x x   + 28x x   + 37x x   + 9x x   + 27x x   - 25x x   + 9x x   + 27x x   - 27x x  ,
                1 3     2 3      3      2 4      3 4    0 5      2 5      3 5    0 6      2 6      1 7      3 7      5 7     6 7     0 8      1 8      3 8      4 8      6 8     0 9      1 9      2 9     3 9     4 9      5 9      6 9     0 10      1 10      2 10     4 10      6 10      0 11     2 11      4 11      5 11
               
                2               2
               x  + 17x x  - 14x  - 13x x  + 34x x  + 44x x  - 30x x  + 27x x  + 31x x  - 36x x  - x x  + 13x x  + 8x x  + 9x x  + 46x x  + 41x x  - 7x x  - 34x x  - 9x x  - 46x x  - 17x x  + 32x x  - 8x x  - 35x x  - 46x x  + 26x x  + 17x x  + 15x x   + 35x x   + 34x x   + 20x x   + 14x x   + 36x x   + 35x x   - 17x x  ,
                2      2 3      3      2 4      3 4      0 5      2 5      3 5      2 6      3 6    0 7      1 7     3 7     5 7      6 7      0 8     1 8      3 8     4 8      6 8      0 9      1 9     2 9      3 9      4 9      5 9      6 9      0 10      1 10      2 10      4 10      0 11      1 11      2 11      4 11
               
                                  2
               x x  - 40x x  + 28x  - x x  + 5x x  - 16x x  + 5x x  - 36x x  + 37x x  + 48x x  - 5x x  - 5x x  + x x  + 20x x  + 10x x  + 34x x  + 41x x  - x x  + x x  + 40x x  - 32x x  + 5x x  - 11x x  - 20x x  + 45x x  - 14x x  - 25x x   + 45x x   - 41x x   - 46x x   + 8x x   - 28x x   + 11x x   + 14x x   - 8x x
                1 2      2 3      3    0 4     2 4      3 4     0 5      2 5      3 5      2 6     1 7     3 7    5 7      6 7      0 8      1 8      3 8    4 8    6 8      0 9      1 9     2 9      3 9      4 9      5 9      6 9      0 10      1 10      2 10      4 10     6 10      0 11      2 11      4 11     5 11
              }
                   ZZ
      target: Proj(--[t , t , t , t , t , t , t , t , t ])
                   97  0   1   2   3   4   5   6   7   8
      defining forms: {
                                   2
                       - 2x x  + 6x  - 24x x  - 25x x  + 39x x  + 3x x  + 46x x  - 39x x  - 17x x  - 38x x  + 24x x  + 24x x  + 5x x  + x x  - 19x x  - 37x x  + 12x x  - 5x x  - 5x x  + 21x x  + 42x x  - 5x x  + x x  - x x  + 2x x  + 48x x  - x x   + 26x x   - 12x x   - 36x x   - 18x x   - 5x x   - 37x x   - 5x x   - 9x x   + 19x x  ,
                           2 3     3      2 4      3 4      0 5     2 5      3 5      0 6      2 6      3 6      1 7      3 7     5 7    6 7      0 8      1 8      3 8     4 8     6 8      0 9      1 9     2 9    3 9    4 9     5 9      6 9    0 10      1 10      2 10      4 10      6 10     0 11      2 11     3 11     4 11      5 11
                       
                       - 19x x  - 40x x  + 19x x  + 19x x  - 38x x  - 39x x  - 37x x  - 21x x  + x x   + 2x x   - 5x x   + x x   - 5x x   + 5x x  ,
                            2 5      3 5      1 8      2 8      3 8      5 8      6 8      5 9    1 10     2 10     3 10    5 10     6 10     5 11
                       
                       - 39x x  - 39x x  - 38x x  - 39x x  + 19x x  + 2x x  + 19x x  + 19x x  + 21x x  + 19x x  - 19x x  + x x  - 39x x  + 37x x  + 40x x  + 40x x  - 5x x  + 39x x  - 37x x  + x x   - x x   - x x   + x x   + 5x x   - 5x x  ,
                            2 3      0 4      3 4      0 5      2 5     3 5      0 7      2 7      3 7      0 8      1 8    3 8      4 8      6 8      0 9      2 9     3 9      4 9      5 9    0 10    1 10    2 10    4 10     6 10     5 11
                       
                            2              2
                       - 39x  - 8x x  + 25x  + 19x x  + 46x x  - 22x x  + x x  - 8x x  + 32x x  - 39x x  + x x  - 38x x  - 30x x  - 5x x  + 39x x  + x x  - 27x x  - 27x x  - 42x x  + 11x x  + 2x x  + 22x x  + 34x x  + 42x x  + 14x x  + 8x x  + 16x x  + 27x x  + 24x x  - 11x x  - 6x x  + 2x x  - 5x x   - 10x x   - 34x x   - 8x x   + 15x x   - 25x x   - 24x x   - 2x x   - 15x x  ,
                            1     2 3      3      0 4      2 4      3 4    0 5     2 5      3 5      4 5    0 6      1 6      2 6     3 6      4 6    5 6      1 7      3 7      5 7      6 7     0 8      1 8      3 8      4 8      6 8     0 9      1 9      2 9      3 9      4 9     5 9     6 9     0 10      1 10      2 10     4 10      6 10      0 11      2 11     4 11      5 11
                       
                                                             2
                       - 39x x  + 19x x  + x x  + 8x x  - 25x  - 23x x  - 30x x  - 21x x  - 10x x  - 34x x  - 38x x  - 7x x  + 5x x  + 19x x  + 23x x  + 21x x  + 44x x  - 14x x  - 22x x  - 5x x  - 34x x  - 44x x  + 10x x  - 44x x  - 21x x  - 21x x  - 9x x  + 14x x  - 22x x  + 45x x  + 5x x   + 20x x   + 34x x   + 12x x   - 25x x   + 20x x   - 5x x   + 9x x   - 45x x   + 25x x  ,
                            0 1      0 2    0 3     2 3      3      2 4      3 4      0 5      2 5      3 5      0 6     2 6     3 6      0 7      1 7      3 7      5 7      6 7      0 8     1 8      3 8      4 8      6 8      0 9      1 9      2 9     3 9      4 9      5 9      6 9     0 10      1 10      2 10      4 10      6 10      0 11     1 11     2 11      4 11      5 11
                       
                       - 24x x  - 25x x  + 22x x  + 48x x  - 17x x  + 24x x  + 24x x  + 5x x  + x x  + 41x x  + 12x x  - 5x x  - 26x x  + 42x x  - 24x x  + 37x x  - x x  - 35x x  + 9x x  + 25x x   - 12x x   - 36x x   - 14x x   - 37x x   - 9x x   + 14x x  ,
                            2 4      3 4      2 5      3 5      2 6      1 7      3 7     5 7    6 7      1 8      3 8     4 8      6 8      1 9      2 9      3 9    4 9      5 9     6 9      1 10      2 10      4 10      6 10      2 11     4 11      5 11
                       
                       19x x  + x x  + 19x x  + 39x x  - 19x x  + 37x x  - 36x x  - x x   + 5x x   - 5x x  ,
                          2 5    3 5      5 7      1 8      4 8      6 8      5 9    4 10     6 10     5 11
                       
                       39x x  - 19x x  - 2x x  + 39x x  + 19x x  - x x  - 37x x  + 37x x  + x x   + x x   - 5x x   + 5x x  ,
                          0 5      2 5     3 5      0 8      1 8    3 8      6 8      5 9    1 10    2 10     6 10     5 11
                       
                       43x x  - 30x x  - 19x x  - 40x x  + 44x x  - 22x x  - 5x x  - 43x x  - 22x x  - 43x x  - 3x x  - 19x x  - 42x x  - 13x x  + 43x x  - 26x x  - 45x x  + 22x x  - 42x x  + 3x x  - 13x x  - 4x x  - 24x x   + 13x x   + 39x x   - 6x x   + 5x x   + 42x x   + 4x x   + 6x x
                          2 4      3 4      0 5      2 5      3 5      2 6     3 6      1 7      3 7      5 7     6 7      0 8      1 8      3 8      4 8      6 8      1 9      2 9      3 9     4 9      5 9     6 9      1 10      2 10      4 10     6 10     1 11      2 11     4 11     5 11
                      }

o10 : RationalMap (quadratic birational map from 8-dimensional subvariety of PP^11 to PP^8)
i11 : assert(phi * psi == 1)

The method also accepts as input a ring map representing a rational map between projective varieties. In this case, a ring map is returned as well.

Caveat

For the purpose of this method, the option MathMode=>true is too rigid, especially when the source of the passed map is not a projective space.

See also

Ways to use approximateInverseMap :

For the programmer

The object approximateInverseMap is a method function with options.