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Cremona :: isDominant

isDominant -- whether a rational map is dominant

Synopsis

Description

This method is based on the fibre dimension theorem. A more general way is to perform the command kernel phi == 0.

i1 : P8=ZZ/101[x_0..x_8];
i2 : phi=toMap ideal jacobian ideal det matrix{{x_0..x_4},{x_1..x_5},{x_2..x_6},{x_3..x_7},{x_4..x_8}}

                 4       2      2 2         2      3     2           2                              2 2        2    3                    2      2                       3     2           2           2         2       3     3         2       2                                     2         2       2       2                       2                              2 2       3     3                     2      2 2         2         2      3       2       2                       2                                           2 2         2        2     2           2                                2       2                                 3           2       3       2       2                       2           2         2         2     2           2                     2       2                                     2         2     3                     2       2                                                        4        2       2 2         2       3     2           2                                 2       2 2         2         2     3                     2       2                                            2 2        2       2       2                       2                              2                       3     2           2           2         2       3     3                     2       2                                     2         2       2       2           2                               2                     2           2                               2                    2 2       3     3                     2      2 2         2         2       2       2                       2                               2 2         2     2           2                               2                    3                    2      2                     3           2       3       2       2           2           2         2     2           2                               2                     3                     2       2                   4       2      2 2         2      3     2           2                              2 2        2    3                    2      2
o2 = map(P8,P8,{x  - 3x x x  + x x  + 2x x x  - x x  + 2x x x  - 2x x x  - 2x x x x  + 2x x x x  + x x  - x x x  - x x  + 2x x x x  - x x x  - x x x  + x x x x , - 2x x  + 4x x x  + 2x x x  - 4x x x  - 2x x x  + 2x x  - 2x x  + 2x x x  + 2x x x  + 2x x x x  - 4x x x x  - 2x x x  + 2x x x  + 2x x x  - 2x x x  - 2x x x x  + 2x x x  + 2x x x x  - 2x x x x , 3x x  - 2x x  - 3x x  - 2x x x x  + 2x x x  + x x  + 4x x x  - 2x x x  - x x  + 4x x x  + 2x x x  - 8x x x x  + 2x x x  - 4x x x x  + 2x x x x  + 2x x x x  + 3x x  - 2x x x  - x x x  - 3x x x  + 2x x x  + 4x x x x  - 2x x x x  - x x x  - 3x x x  + 2x x x x  + x x x x , - 4x x  + 6x x x  - 2x x  + 6x x x  - 6x x x  - 2x x x x  + 2x x x  + 2x x x  - 4x x x  + 2x x x  - 6x x x  + 4x x x  + 4x x x x  - 2x x x  - 2x x x  + 4x x x x  - 2x x x x  - 2x x x  + 2x x x  + 4x x  - 6x x x x  + 2x x x  + 2x x x  - 4x x x x  + 2x x x x  + 2x x x x  - 2x x x x , 5x  - 12x x x  + 3x x  + 6x x x  - 2x x  + 6x x x  - 9x x x  - 2x x x x  + 8x x x x  - 3x x x  + 2x x  - 4x x x  + 2x x x  - 2x x  + 8x x x x  - 6x x x  - 4x x x  + 4x x x x  + 2x x x x  - 2x x x x  + x x  - x x x  - 3x x x  + 2x x x  + 4x x x x  - 3x x x  - 2x x x x  + 2x x x x  - x x x  + x x x x , - 4x x  + 6x x x  + 6x x x  - 6x x x  - 6x x x  + 4x x  - 2x x  - 2x x x x  + 4x x x  + 2x x x  + 4x x x x  - 6x x x x  - 2x x x  + 2x x x  + 2x x x  - 4x x x  + 2x x x  + 4x x x x  - 4x x x x  - 2x x x  + 2x x x x  + 2x x x  - 2x x x  - 2x x x x  + 2x x x x  + 2x x x  - 2x x x x , 3x x  - 3x x  - 2x x  - 2x x x x  + 4x x x  + x x  + 2x x x  - 3x x x  + 2x x x  + 4x x x  - 8x x x x  + 2x x x  - 4x x x x  + 4x x x x  + 3x x  - 3x x x  - 2x x x  + 2x x x  + 2x x x x  - 2x x x x  - 2x x x  + 2x x x x  - x x  + 2x x x x  - x x x  - x x x  + x x x x , - 2x x  + 4x x x  - 2x x  + 2x x x  - 4x x x  + 2x x x  + 2x x x  - 2x x x  - 2x x x  + 2x x x  + 2x x x x  - 2x x x x  - 2x x x  + 2x x x x  + 2x x  - 4x x x x  + 2x x x  + 2x x x  - 2x x x x , x  - 3x x x  + x x  + 2x x x  - x x  + 2x x x  - 2x x x  - 2x x x x  + 2x x x x  + x x  - x x x  - x x  + 2x x x x  - x x x  - x x x  + x x x x })
                 5     4 5 6    4 6     3 5 6    2 6     4 5 7     3 5 7     3 4 6 7     2 5 6 7    3 7    2 4 7    4 8     3 4 5 8    2 5 8    3 6 8    2 4 6 8      4 5     4 5 6     3 5 6     3 4 6     2 5 6     1 6     4 7     2 5 7     3 6 7     2 4 6 7     1 5 6 7     2 3 7     1 4 7     3 4 8     3 5 8     2 4 5 8     1 5 8     2 3 6 8     1 4 6 8    4 5     3 5     4 6     3 4 5 6     2 5 6    3 6     2 4 6     1 5 6    0 6     3 4 7     3 5 7     2 4 5 7     1 5 7     2 3 6 7     1 4 6 7     0 5 6 7     2 7     1 3 7    0 4 7     3 4 8     2 4 8     2 3 5 8     1 4 5 8    0 5 8     2 6 8     1 3 6 8    0 4 6 8      4 5     3 4 5     2 5     3 4 6     3 5 6     2 4 5 6     1 5 6     2 3 6     1 4 6     0 5 6     3 4 7     2 4 7     2 3 5 7     0 5 7     2 6 7     1 3 6 7     0 4 6 7     1 2 7     0 3 7     3 8     2 3 4 8     1 4 8     2 5 8     1 3 5 8     0 4 5 8     1 2 6 8     0 3 6 8    4      3 4 5     3 5     2 4 5     1 5     3 4 6     2 4 6     2 3 5 6     1 4 5 6     0 5 6     2 6     1 3 6     0 4 6     3 7     2 3 4 7     1 4 7     2 5 7     0 4 5 7     1 2 6 7     0 3 6 7    1 7    0 2 7     2 3 8     2 4 8     1 3 4 8     0 4 8     1 2 5 8     0 3 5 8    1 6 8    0 2 6 8      3 4     3 4 5     2 4 5     2 3 5     1 4 5     0 5     3 6     2 3 4 6     1 4 6     2 5 6     1 3 5 6     0 4 5 6     1 2 6     0 3 6     2 3 7     2 4 7     0 4 7     1 2 5 7     0 3 5 7     1 6 7     0 2 6 7     2 3 8     1 3 8     1 2 4 8     0 3 4 8     1 5 8     0 2 5 8    3 4     2 4     3 5     2 3 4 5     1 4 5    2 5     1 3 5     0 4 5     2 3 6     2 4 6     1 3 4 6     0 4 6     1 2 5 6     0 3 5 6     1 6     0 2 6     2 3 7     1 3 7     1 2 4 7     0 3 4 7     1 5 7     0 2 5 7    2 8     1 2 3 8    0 3 8    1 4 8    0 2 4 8      3 4     2 3 4     1 4     2 3 5     2 4 5     0 4 5     1 2 5     0 3 5     2 3 6     1 3 6     1 2 4 6     0 3 4 6     1 5 6     0 2 5 6     2 7     1 2 3 7     0 3 7     1 4 7     0 2 4 7   3     2 3 4    2 4     1 3 4    0 4     2 3 5     1 3 5     1 2 4 5     0 3 4 5    1 5    0 2 5    2 6     1 2 3 6    0 3 6    1 4 6    0 2 4 6

o2 : RingMap P8 <--- P8
i3 : time isDominant(phi,MathMode=>true)
MathMode: output certified!
     -- used 2.86901 seconds

o3 = true
i4 : P7=ZZ/101[x_0..x_7];
i5 : -- hyperelliptic curve of genus 3
     C=ideal(x_4*x_5+23*x_5^2-23*x_0*x_6-18*x_1*x_6+6*x_2*x_6+37*x_3*x_6+23*x_4*x_6-26*x_5*x_6+2*x_6^2-25*x_0*x_7+45*x_1*x_7+30*x_2*x_7-49*x_3*x_7-49*x_4*x_7+50*x_5*x_7,x_3*x_5-24*x_5^2+21*x_0*x_6+x_1*x_6+46*x_3*x_6+27*x_4*x_6+5*x_5*x_6+35*x_6^2+20*x_0*x_7-23*x_1*x_7+8*x_2*x_7-22*x_3*x_7+20*x_4*x_7-15*x_5*x_7,x_2*x_5+47*x_5^2-40*x_0*x_6+37*x_1*x_6-25*x_2*x_6-22*x_3*x_6-8*x_4*x_6+27*x_5*x_6+15*x_6^2-23*x_0*x_7-42*x_1*x_7+27*x_2*x_7+35*x_3*x_7+39*x_4*x_7+24*x_5*x_7,x_1*x_5+15*x_5^2+49*x_0*x_6+8*x_1*x_6-31*x_2*x_6+9*x_3*x_6+38*x_4*x_6-36*x_5*x_6-30*x_6^2-33*x_0*x_7+26*x_1*x_7+32*x_2*x_7+27*x_3*x_7+6*x_4*x_7+36*x_5*x_7,x_0*x_5+30*x_5^2-11*x_0*x_6-38*x_1*x_6+13*x_2*x_6-32*x_3*x_6-30*x_4*x_6+4*x_5*x_6-28*x_6^2-30*x_0*x_7-6*x_1*x_7-45*x_2*x_7+34*x_3*x_7+20*x_4*x_7+48*x_5*x_7,x_3*x_4+46*x_5^2-37*x_0*x_6+27*x_1*x_6+33*x_2*x_6+8*x_3*x_6-32*x_4*x_6+42*x_5*x_6-34*x_6^2-37*x_0*x_7-28*x_1*x_7+10*x_2*x_7-27*x_3*x_7-42*x_4*x_7-8*x_5*x_7,x_2*x_4-25*x_5^2-4*x_0*x_6+2*x_1*x_6-31*x_2*x_6-5*x_3*x_6+16*x_4*x_6-24*x_5*x_6+31*x_6^2-30*x_0*x_7+32*x_1*x_7+12*x_2*x_7-40*x_3*x_7+3*x_4*x_7-28*x_5*x_7,x_0*x_4+15*x_5^2+48*x_0*x_6-50*x_1*x_6+46*x_2*x_6-48*x_3*x_6-23*x_4*x_6-28*x_5*x_6+39*x_6^2+38*x_1*x_7-5*x_3*x_7+5*x_4*x_7-34*x_5*x_7,x_3^2-31*x_5^2+41*x_0*x_6-30*x_1*x_6-4*x_2*x_6+43*x_3*x_6+23*x_4*x_6+7*x_5*x_6+31*x_6^2-19*x_0*x_7+25*x_1*x_7-49*x_2*x_7-16*x_3*x_7-45*x_4*x_7+25*x_5*x_7,x_2*x_3+13*x_5^2-45*x_0*x_6-22*x_1*x_6+33*x_2*x_6-26*x_3*x_6-21*x_4*x_6+34*x_5*x_6-21*x_6^2-47*x_0*x_7-10*x_1*x_7+29*x_2*x_7-46*x_3*x_7-x_4*x_7+20*x_5*x_7,x_1*x_3+22*x_5^2+4*x_0*x_6+3*x_1*x_6+45*x_2*x_6+37*x_3*x_6+17*x_4*x_6+36*x_5*x_6-2*x_6^2-31*x_0*x_7+3*x_1*x_7-12*x_2*x_7+19*x_3*x_7+28*x_4*x_7+30*x_5*x_7,x_0*x_3-47*x_5^2-43*x_0*x_6+6*x_1*x_6-40*x_2*x_6+21*x_3*x_6+26*x_4*x_6-5*x_5*x_6-5*x_6^2+4*x_0*x_7-15*x_1*x_7+18*x_2*x_7-31*x_3*x_7+50*x_4*x_7-46*x_5*x_7,x_2^2+4*x_5^2+31*x_0*x_6+41*x_1*x_6+31*x_2*x_6+28*x_3*x_6+42*x_4*x_6-28*x_5*x_6-4*x_6^2-7*x_0*x_7+15*x_1*x_7-9*x_2*x_7+31*x_3*x_7+3*x_4*x_7+7*x_5*x_7,x_1*x_2-46*x_5^2-6*x_0*x_6-50*x_1*x_6+32*x_2*x_6-10*x_3*x_6+42*x_4*x_6+33*x_5*x_6+18*x_6^2-9*x_0*x_7-20*x_1*x_7+45*x_2*x_7-9*x_3*x_7+10*x_4*x_7-8*x_5*x_7,x_0*x_2-9*x_5^2+34*x_0*x_6-45*x_1*x_6+19*x_2*x_6+24*x_3*x_6+23*x_4*x_6-37*x_5*x_6-44*x_6^2+24*x_0*x_7-33*x_2*x_7+41*x_3*x_7-40*x_4*x_7+4*x_5*x_7,x_1^2+x_1*x_4+x_4^2-28*x_5^2-33*x_0*x_6-17*x_1*x_6+11*x_3*x_6+20*x_4*x_6+25*x_5*x_6-21*x_6^2-22*x_0*x_7+24*x_1*x_7-14*x_2*x_7+5*x_3*x_7-39*x_4*x_7-18*x_5*x_7,x_0*x_1-47*x_5^2-5*x_0*x_6-9*x_1*x_6-45*x_2*x_6+48*x_3*x_6+45*x_4*x_6-29*x_5*x_6+3*x_6^2+29*x_0*x_7+40*x_1*x_7+46*x_2*x_7+27*x_3*x_7-36*x_4*x_7-39*x_5*x_7,x_0^2-31*x_5^2+36*x_0*x_6-30*x_1*x_6-10*x_2*x_6+42*x_3*x_6+9*x_4*x_6+34*x_5*x_6-6*x_6^2+48*x_0*x_7-47*x_1*x_7-19*x_2*x_7+25*x_3*x_7+28*x_4*x_7+34*x_5*x_7);

o5 : Ideal of P7
i6 : phi=toMap(C,3,2)

                     2                 2          2        2      3                 2                            2          2        2        2                                       2                                         2          2        2        2                 2                              2                                         2                                                    2         2        2        2        2        2      3                 2                              2                                         2                                                    2                   2                             2                                                                                     2                                                               2           2        2     3                2                              2          2        2        2                2                                        2         2                 2                             2                                         2                                                   2          2        2       2        2        2     3      2                              2                                         2                                                    2                2                  2                                                                               2                                                           2       3     2          2    2                   2          2        2      3                 2                              2          2       2        2     3                                      2                                         2          2        2        2                 2                             2                                        2                                                    2          2        2        2        2        2      3      2                   2                              2                                       2                                                   2                                                               2          2        2       2        2        2      2                  2                              2                                         2                                                  2                                                              2        3      2          2                 2          2        2      3    2                  2                              2          2        2        2      3                                      2                                         2          2        2        2                 2                             2                                       2                                                    2          2        2        2        2        2      3      2                   2                              2                                         2                                                    2                                                               2          2        2        2       2       2      2                   2                             2                                        2                                                   2                                                             2         3      2          2                 2          2       2      3                 2                             2         2       2       2     3    2                                         2                                        2          2        2        2                 2                              2                                         2                                                    2          2       2        2        2        2      3      2                   2                             2                                         2                                                    2                                                            2          2        2        2        2        2      2                  2                              2                                         2                                                    2                                                              2         3      2        2                2          2       2      3                2                            2          2        2        2      3                                    2                                         2        2       2        2    2                  2                             2                                         2                                                  2        2        2        2        2        2     3      2                  2                             2                                        2                                                 2                                                               2          2        2        2        2      2                   2                              2                                        2                                                   2                                                              2         3      2         2                 2          2        2      3                2                              2          2        2      3                                      2                                       2         2        2       2                 2                              2                                        2                                                    2         2        2        2      2        2      3     2                   2                              2                                      2                                                  2                                                             2          2        2        2        2        2     2                   2                           2                                        2                                                    2                                                              2         3      2          2                 2          2        2      3                 2                              2          2       2        2      3                                                                        2          2        2        2                 2                             2                                        2                                         2          2        2        2        2        2      3      2                   2                             2                                        2                                                  2                                                               2          2        2       2        2        2      2                   2                             2                                         2                                                   2                                                              2
o6 = map(P7,P7,{17x x  + 44x x x  - 34x x  + 46x x  + 10x x  - 17x  + 29x x x  - 47x x  - 20x x x  - 44x x x  - x x  - 27x x  - 46x x  + 31x x  + 17x x x  + 13x x x  - 34x x x  - 37x x  + 30x x x  - 47x x x  - 45x x x  - 27x x  + 17x x  - 34x x  - 47x x  + 23x x x  + 31x x  - 30x x x  - 24x x x  + 43x x  + 18x x x  + 26x x x  + 33x x x  - 27x x  + 19x x x  + 31x x x  + 11x x x  - 29x x x  + 31x x  + 5x x  + 40x x  - 47x x  + 18x x  - 31x x  + 39x  + 17x x x  + 41x x  - 18x x x  + 27x x x  - 19x x  + 41x x x  + 41x x x  - 18x x x  + 41x x  - 27x x x  - 29x x x  + 13x x x  + 14x x x  - 27x x  - 12x x x  - 30x x  - 9x x x  + 50x x x  - 21x x  + 33x x x  + 31x x x  + 31x x x  + 47x x x  - 30x x x  - 22x x x  + 33x x x  - 30x x  + 47x x x  - 30x x x  - 22x x x  + 33x x x  + 41x x x  - 30x x , - 41x x  - 29x x  - 4x  - 10x x x  - 6x x  + 30x x x  - 34x x x  - 14x x  - 10x x  + 43x x  - 10x x  + 11x x x  - 6x x  + 42x x x  - 6x x x  - 10x x x  - 10x x  - 6x x  - 41x x x  - 29x x  - 31x x x  - 27x x x  - 4x x  - 41x x x  + 37x x x  + 12x x x  - 10x x  - 6x x x  - 29x x x  - 16x x x  - 47x x x  - 29x x  + 40x x  - 17x x  - 4x x  - 41x x  - 33x x  - 4x  - 11x x  + 41x x x  + 43x x x  - 26x x  - 11x x x  - 11x x x  - 23x x x  - 11x x  - 10x x x  + 43x x x  + 31x x x  - 21x x x  - 10x x  + x x x  - 6x x  + 47x x x  + 6x x  + x x x  - 47x x x  - 47x x x  + 42x x x  - 6x x x  + 19x x x  + x x x  - 6x x  + 42x x x  - 6x x x  + 19x x x  + x x x  - 12x x x  - 6x x , 39x  + 4x x  + 33x x  + x x  + 12x x x  + 39x x  + 48x x  + 22x x  + 29x  - 27x x x  - 36x x  - 50x x x  - 36x x x  + 35x x  + 35x x  + 8x x  + 34x x  + 2x  + 33x x x  + 18x x x  + 39x x x  + 7x x  + 32x x x  - 36x x x  + 40x x x  - 10x x  + 33x x  + 39x x  - 36x x  + 24x x x  - 16x x  - 46x x x  - 8x x x  + 31x x  - 21x x x  - 5x x x  - 15x x x  - 43x x  - 40x x x  - 16x x x  - 39x x x  + 39x x x  - 16x x  - 43x x  + 18x x  + 26x x  - 30x x  - 22x x  - 37x  - 10x x  + 21x x x  - 13x x  - 42x x x  - 31x x x  - 31x x  + 11x x x  - 38x x x  - 3x x x  - 4x x  + 42x x x  - 13x x x  + 5x x x  - 20x x x  - 13x x  - 46x x x  + 20x x x  + 42x x x  + 19x x x  + 18x x x  + 31x x  + 19x x  - 33x x  - 2x x  + 15x x  + 15x x  - 47x x  + 23x x x  - 3x x  + 42x x x  - 35x x x  + 36x x  + 13x x x  + 33x x x  + 37x x x  - 31x x  + 22x x x  - 3x x x  + 27x x x  + 44x x x  - 3x x  + 3x x x  + 30x x x  + 29x x x  + 44x x x  - 21x x x  - 18x x , - 9x  - 21x x  + 15x x  + 21x x x  - 16x x  + 39x x  + 13x x  + 40x  + x x  + 9x x x  + 34x x  - 31x x x  + 36x x x  + 14x x  + 10x x  + 17x x  + 45x x  - 31x  + 15x x x  - 9x x x  - 16x x x  - 38x x  + 17x x x  + 34x x x  - 13x x x  - 10x x  + 15x x  - 16x x  + 34x x  + 19x x x  - 43x x  - 10x x x  + 5x x x  - 35x x  + 8x x x  + 26x x x  - 2x x x  + 29x x  - 13x x x  - 43x x x  - 37x x x  - 38x x x  - 43x x  - 10x x  - 36x x  + 49x x  - 27x x  - 18x x  - 13x  - 21x x  + 45x x x  - 44x x  - 11x x x  + 24x x x  + 18x x  - 33x x x  - 29x x x  - 44x x x  + 45x x  - 14x x x  - 44x x x  - 27x x x  + 37x x x  - 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3 7      0 4 7      1 4 7      2 4 7      3 4 7      4 7      0 5 7      1 5 7     2 5 7      3 5 7      4 5 7      5 7       0      0 1      0 1      0 1 2      1 2      0 2      1 2      2      0 1 3      1 3      0 2 3      1 2 3      2 3      0 3     1 3      2 3      3      0 1 4      0 2 4      1 2 4      0 3 4      1 3 4      2 3 4      3 4      0 4      2 4      3 4      0 1 5      1 5      0 2 5     1 2 5      2 5      0 3 5      1 3 5     2 3 5      3 5      0 4 5      1 4 5      3 4 5      4 5      0 5      1 5      2 5      3 5      4 5      5      0 6      0 1 6      1 6      0 2 6      1 2 6     2 6      0 3 6      1 3 6      2 3 6     3 6     0 4 6      1 4 6      2 4 6     3 4 6      4 6      0 5 6      1 5 6      2 5 6      3 5 6      4 5 6      5 6      0 6      1 6     2 6      4 6      5 6      0 7      0 1 7      1 7     0 2 7      1 2 7      2 7      0 3 7      1 3 7      2 3 7      3 7     0 4 7      1 4 7      2 4 7      3 4 7      4 7      0 5 7      1 5 7      2 5 7      3 5 7     4 5 7      5 7

o6 : RingMap P7 <--- P7
i7 : time isDominant(phi,MathMode=>true)
MathMode: output certified!
     -- used 2.5303 seconds

o7 = false

Ways to use isDominant :