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Cremona :: degreeOfRationalMap

degreeOfRationalMap -- degree of a rational map with source a projective space

Synopsis

Description

Let φ:K[y0,...,ym]/J →K[x0,...,xn] be a ring map representing a rational map Φ:ℙn=Proj(K[x0,...,xn]) --->V(J)⊆ℙm=Proj(K[y0,...,ym]). If p is a general point of n, denote by Fp(Φ) the closure of Φ-1(Φ(p))⊆ℙn. The degree of Φ is defined as the degree of Fp(Φ) if dim Fp(Φ) = 0 and 0 otherwise. If Φ is defined by forms F0(x0,...,xn),...,Fm(x0,...,xn) and Ip is the ideal of the point p, then the ideal of Fp(Φ) is nothing but the saturation (φ(φ-1(Ip))):(F0,....,Fm).
i1 : -- Take a birational map phi:P^8--->G(1,5) subset P^14 defined by the maximal minors 
     -- of a generic 2 x 6 matrix of linear forms on P^8 (thus phi is birational onto its image)
     K=ZZ/331; ringP8=K[x_0..x_8]; ringP14=K[t_0..t_14];
i4 : phi=map(ringP8,ringP14,gens minors(2,matrix pack(6,for i to 11 list random(1,ringP8))))

                              2                2                          2                                   2                                           2                                                   2                                                             2                                                                         2                                                                                  2     2               2                         2                                  2                                              2                                                      2                                                              2                                                                          2                                                                                   2       2                 2                          2                                  2                                           2                                    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                    2
o4 = map(ringP8,ringP14,{- 37x  - 163x x  + 64x  - 153x x  - 147x x  + 91x  - 159x x  + 70x x  + 133x x  + 62x  + 111x x  - 24x x  - 129x x  + 62x x  + 7x  + 11x x  - 54x x  + 7x x  + 95x x  + 105x x  - 63x  - 30x x  + 65x x  + 145x x  + 13x x  - 99x x  + 29x x  + 92x  + 89x x  + 51x x  - 137x x  - 111x x  + 126x x  - 127x x  - 80x x  - 28x  - 19x x  - 27x x  - 59x x  + 141x x  + 24x x  + 66x x  + 107x x  + 100x x  - 116x , 67x  + 26x x  - 96x  + 110x x  - 11x x  - 41x  + 46x x  - 142x x  + 87x x  + 40x  - 160x x  - 159x x  + 73x x  - 130x x  + 123x  + 114x x  + 18x x  - 124x x  - 152x x  - 26x x  + 12x  + 52x x  - 40x x  - 120x x  + 62x x  + 84x x  + 58x x  + 122x  + 152x x  - 23x x  - 110x x  + 120x x  + 157x x  + 8x x  + 141x x  - 108x  - 74x x  - 155x x  + 130x x  + 65x x  - 127x x  + 40x x  - 116x x  + 159x x  - 89x , - 21x  + 116x x  + 133x  + 155x x  + 79x x  - 148x  - 61x x  - 115x x  - 70x x  + 25x  + 86x x  - 68x x  - 41x x  + 145x x  - 11x  + 137x x  - 6x x  - 114x x  - 26x x  + 114x x  + 3x  - 84x x  - 17x x  - 160x x  - 32x x  + 46x x  + 132x x  - 153x  - 109x x  + 160x x  - 74x x  + 114x x  - 114x x  - 109x x  - 65x x  - 7x  - 120x x  - 151x x  + 157x x  + 21x x  - 14x x  - 40x x  - 71x x  + 132x x  - 21x , 106x  + 34x x  - 118x  + 143x x  + 120x x  - 27x  - 114x x  - 162x x  + 67x x  + 144x  + 106x x  + 83x x  - 62x x  - 36x x  - 102x  + 160x x  + 147x x  + 42x x  + 12x x  - 65x x  + 52x  + 108x x  - 37x x  - 127x x  - 90x x  + 84x x  - 6x x  - 33x  + 160x x  + 14x x  - 10x x  + 114x x  + 84x x  + 86x x  - 37x x  + 66x  + 104x x  - 83x x  - 71x x  - 149x x  - 131x x  + 19x x  - 143x x  + 61x x  - 164x , - 33x  + 37x x  + 35x  - 152x x  - 55x x  + 130x  - 23x x  + 81x x  - 136x x  + 161x  + 57x x  - 77x x  - 5x x  + 47x x  - 120x  - 106x x  - 150x x  + 86x x  + 89x x  + 160x x  - 115x  + 43x x  - 143x x  + 16x x  + 150x x  + 75x x  + 106x x  + 15x  + 165x x  + 154x x  + 112x x  + 23x x  + 134x x  - 107x x  + 153x x  - 151x  + 85x x  - 33x x  - 2x x  - 29x x  - 70x x  - 44x x  - 125x x  - 11x x  - 148x , 98x  - 82x x  + 141x  + 7x x  + 3x  - 37x x  - 146x x  - 98x x  - 93x  + 118x x  - 113x x  + 96x x  - 24x x  - 141x  + 34x x  - 61x x  + 153x x  + 101x x  + 122x x  - 149x  - 64x x  + 126x x  - 161x x  - 162x x  + 10x x  + 110x x  - 17x  + 72x x  - 59x x  - 114x x  - 28x x  + 10x x  - 50x x  - 122x x  + 134x  - 20x x  + 120x x  - 4x x  + 154x x  + 75x x  - 28x x  - 72x x  + 100x x  + 133x , 64x  - 127x x  + 51x  + 67x x  + 126x x  - 42x  + 9x x  - 2x x  - 150x x  - 141x  + 69x x  + 10x x  + 35x x  + 126x x  + 65x  + 127x x  + 47x x  - 102x x  + 53x x  - 36x x  + 77x  - 51x x  - 12x x  - 102x x  + 4x x  - 124x x  - 15x x  - 26x  + 72x x  - 14x x  - 137x x  - 12x x  - 86x x  - 24x x  - 103x x  + 58x  + 141x x  + 68x x  - 156x x  + 42x x  + 82x x  - 93x x  + 163x x  + 53x x  + 2x , - 30x  - 122x x  + 151x  + 26x x  - 43x x  + 63x  + 40x x  + 40x x  + 38x x  - 38x  + 144x x  - 83x x  - 62x x  + 151x x  + 31x  + 27x x  + 53x x  - 148x x  + 57x x  - 6x x  + 134x  - 92x x  + 136x x  - 86x x  - 66x x  + 146x x  - 92x x  + 104x  + 120x x  + 10x x  - 161x x  - 159x x  + 148x x  - 30x x  + 34x x  - 60x  + 119x x  + 32x x  - 47x x  + 71x x  - 76x x  - 75x x  - 110x x  + 49x x  - 144x , 18x  - 155x x  - 17x  - 8x x  + 135x x  + 107x  + 90x x  - 141x x  - 84x x  + 27x  + 122x x  - 115x x  + 39x x  + 50x x  - 157x  - 153x x  - 46x x  + 56x x  - 56x x  - 21x x  + 120x  - 119x x  + 47x x  - 69x x  + 133x x  + 95x x  + 89x x  - 114x  + 24x x  - 30x x  + 46x x  + 163x x  + 47x x  + 38x x  + 56x x  + 156x  + 36x x  - 85x x  - 88x x  - 78x x  - 155x x  - 135x x  + 103x x  - 107x x  + 9x , - 159x  + 65x x  + 154x  + 90x x  + 148x x  - 86x  - 162x x  + 145x x  + 59x  + 137x x  - 34x x  - 126x x  - 40x x  - 94x  + 120x x  - 65x x  - 106x x  + 30x x  + 147x x  - 41x  + 90x x  - 105x x  - 96x x  - 20x x  + 51x x  - 120x x  - 141x  - 98x x  - 37x x  - 34x x  + 145x x  + 26x x  + 137x x  - 113x x  + 136x  - 100x x  - 118x x  - 80x x  - 85x x  - 121x x  + 63x x  + 131x x  + 15x x  - 92x , - 103x  + 14x x  - 104x  - 94x x  - 92x x  + 17x  - 99x x  + 151x x  + 13x x  - 25x  - 84x x  + 85x x  - 120x x  + 32x x  + 46x  - 147x x  + 113x x  + 138x x  - 109x x  - 160x x  - 26x  - 36x x  - 66x x  - 16x x  - 102x x  + 13x x  - 79x x  - 25x  - 22x x  - 89x x  + 14x x  - 93x x  + 18x x  - 133x x  - 154x x  + 51x  + 132x x  - 120x x  - 46x x  - 86x x  + 89x x  - 124x x  + 40x x  - 30x x  + 37x , - 52x  + 26x x  + 5x  + 32x x  + 89x x  - 110x  - 5x x  - 121x x  + 79x x  - 32x  - 154x x  - 48x x  + 81x x  - 86x x  - 47x  + 80x x  + 9x x  - 45x x  + 157x x  - 112x x  - 34x  - 59x x  - 101x x  - 153x x  - 3x x  - 134x x  + 27x x  - 130x  - 157x x  + 74x x  + 147x x  - 160x x  - 164x x  + 54x x  - 67x x  + 155x  - 79x x  - 70x x  + 117x x  - 31x x  - 32x x  - 20x x  - 131x x  - 9x x  + 49x , 90x  - 58x x  - 81x  - 99x x  - 156x x  - 21x  - 96x x  - 71x x  + 26x x  - 112x  - 154x x  + 11x x  + 29x x  - 7x x  + 3x  + 102x x  - 144x x  - 139x x  - 101x x  + 140x x  + 21x  + 66x x  + 4x x  + 84x x  - 89x x  + 152x x  - 138x x  - 160x  + 91x x  - 136x x  + 119x x  - 74x x  + 3x x  - 141x x  - 21x x  - 136x  + 82x x  + 11x x  - 73x x  - 60x x  - 150x x  + 58x x  + 76x x  - 6x x  - 49x , - 117x  - 83x x  + 58x  - 45x x  - 164x x  + 52x  - 117x x  - 16x x  + x x  + 60x  + 57x x  + 152x x  + 6x x  - x x  - 87x  - 96x x  + 121x x  + 40x x  + 137x x  + 52x x  + 44x  - 155x x  - 46x x  + 85x x  + 96x x  + 30x x  - 157x x  - 104x  - 127x x  - 74x x  - 56x x  - 147x x  - 112x x  + 95x x  - 5x x  + 128x  + 114x x  - 53x x  + 140x x  - 153x x  + 165x x  - 131x x  + 46x x  - 35x , 84x  - 140x x  + 58x  - 90x x  + 25x x  + 39x  - 165x x  + 159x x  - 41x x  - 60x  + 50x x  - 46x x  + 143x x  - 108x x  - 120x  + 127x x  + 81x x  + 95x x  - 48x x  + 142x x  - 19x  - 155x x  + 14x x  + 148x x  + 51x x  - 5x x  + 25x x  + 65x  + 26x x  - 150x x  + 35x x  - 56x x  + 115x x  - 101x x  - 49x x  - 123x  - 84x x  - 146x x  + 95x x  - 31x x  - 21x x  - 31x x  - 64x x  + 16x })
                              0       0 1      1       0 2       1 2      2       0 3      1 3       2 3      3       0 4      1 4       2 4      3 4     4      0 5      1 5     2 5      3 5       4 5      5      0 6      1 6       2 6      3 6      4 6      5 6      6      0 7      1 7       2 7       3 7       4 7       5 7      6 7      7      0 8      1 8      2 8       3 8      4 8      5 8       6 8       7 8       8     0      0 1      1       0 2      1 2      2      0 3       1 3      2 3      3       0 4       1 4      2 4       3 4       4       0 5      1 5       2 5       3 5      4 5      5      0 6      1 6       2 6      3 6      4 6      5 6       6       0 7      1 7       2 7       3 7       4 7     5 7       6 7       7      0 8       1 8       2 8      3 8       4 8      5 8       6 8       7 8      8       0       0 1       1       0 2      1 2       2      0 3       1 3      2 3      3      0 4      1 4      2 4       3 4      4       0 5     1 5       2 5      3 5       4 5     5      0 6      1 6       2 6      3 6      4 6       5 6       6       0 7       1 7      2 7       3 7       4 7       5 7      6 7     7       0 8       1 8       2 8      3 8      4 8      5 8      6 8       7 8      8      0      0 1       1       0 2       1 2      2       0 3       1 3      2 3       3       0 4      1 4      2 4      3 4       4       0 5       1 5      2 5      3 5      4 5      5       0 6      1 6       2 6      3 6      4 6     5 6      6       0 7      1 7      2 7       3 7      4 7      5 7      6 7      7       0 8      1 8      2 8       3 8       4 8      5 8       6 8      7 8       8       0      0 1      1       0 2      1 2       2      0 3      1 3       2 3       3      0 4      1 4     2 4      3 4       4       0 5       1 5      2 5      3 5       4 5       5      0 6       1 6      2 6       3 6      4 6       5 6      6       0 7       1 7       2 7      3 7       4 7       5 7       6 7       7      0 8      1 8     2 8      3 8      4 8      5 8       6 8      7 8       8     0      0 1       1     1 2     2      0 3       1 3      2 3      3       0 4       1 4      2 4      3 4       4      0 5      1 5       2 5       3 5       4 5       5      0 6       1 6       2 6       3 6      4 6       5 6      6      0 7      1 7       2 7      3 7      4 7      5 7       6 7       7      0 8       1 8     2 8       3 8      4 8      5 8      6 8       7 8       8     0       0 1      1      0 2       1 2      2     0 3     1 3       2 3       3      0 4      1 4      2 4       3 4      4       0 5      1 5       2 5      3 5      4 5      5      0 6      1 6       2 6     3 6       4 6      5 6      6      0 7      1 7       2 7      3 7      4 7      5 7       6 7      7       0 8      1 8       2 8      3 8      4 8      5 8       6 8      7 8     8       0       0 1       1      0 2      1 2      2      0 3      1 3      2 3      3       0 4      1 4      2 4       3 4      4      0 5      1 5       2 5      3 5     4 5       5      0 6       1 6      2 6      3 6       4 6      5 6       6       0 7      1 7       2 7       3 7       4 7      5 7      6 7      7       0 8      1 8      2 8      3 8      4 8      5 8       6 8      7 8       8     0       0 1      1     0 2       1 2       2      0 3       1 3      2 3      3       0 4       1 4      2 4      3 4       4       0 5      1 5      2 5      3 5      4 5       5       0 6      1 6      2 6       3 6      4 6      5 6       6      0 7      1 7      2 7       3 7      4 7      5 7      6 7       7      0 8      1 8      2 8      3 8       4 8       5 8       6 8       7 8     8        0      0 1       1      0 2       1 2      2       0 3       2 3      3       0 4      1 4       2 4      3 4      4       0 5      1 5       2 5      3 5       4 5      5      0 6       1 6      2 6      3 6      4 6       5 6       6      0 7      1 7      2 7       3 7      4 7       5 7       6 7       7       0 8       1 8      2 8      3 8       4 8      5 8       6 8      7 8      8        0      0 1       1      0 2      1 2      2      0 3       1 3      2 3      3      0 4      1 4       2 4      3 4      4       0 5       1 5       2 5       3 5       4 5      5      0 6      1 6      2 6       3 6      4 6      5 6      6      0 7      1 7      2 7      3 7      4 7       5 7       6 7      7       0 8       1 8      2 8      3 8      4 8       5 8      6 8      7 8      8       0      0 1     1      0 2      1 2       2     0 3       1 3      2 3      3       0 4      1 4      2 4      3 4      4      0 5     1 5      2 5       3 5       4 5      5      0 6       1 6       2 6     3 6       4 6      5 6       6       0 7      1 7       2 7       3 7       4 7      5 7      6 7       7      0 8      1 8       2 8      3 8      4 8      5 8       6 8     7 8      8     0      0 1      1      0 2       1 2      2      0 3      1 3      2 3       3       0 4      1 4      2 4     3 4     4       0 5       1 5       2 5       3 5       4 5      5      0 6     1 6      2 6      3 6       4 6       5 6       6      0 7       1 7       2 7      3 7     4 7       5 7      6 7       7      0 8      1 8      2 8      3 8       4 8      5 8      6 8     7 8      8        0      0 1      1      0 2       1 2      2       0 3      1 3    2 3      3      0 4       1 4     2 4    3 4      4      0 5       1 5      2 5       3 5      4 5      5       0 6      1 6      2 6      3 6      4 6       5 6       6       0 7      1 7      2 7       3 7       4 7      5 7     6 7       7       0 8      1 8       2 8       4 8       5 8       6 8      7 8      8     0       0 1      1      0 2      1 2      2       0 3       1 3      2 3      3      0 4      1 4       2 4       3 4       4       0 5      1 5      2 5      3 5       4 5      5       0 6      1 6       2 6      3 6     4 6      5 6      6      0 7       1 7      2 7      3 7       4 7       5 7      6 7       7      0 8       1 8      2 8      3 8      5 8      6 8      7 8      8

o4 : RingMap ringP8 <--- ringP14
i5 : time degreeOfRationalMap phi
     -- used 0.243662 seconds

o5 = 1
i6 : -- Compose phi:P^8--->P^14 with a linear projection P^14--->P^8 from a general subspace of P^14 
     -- of dimension 5 (so that the composition phi':P^8--->P^8 must have degree equal to deg(G(1,5))=14)
     phi'=phi*map(ringP14,ringP8,for i to 8 list random(1,ringP14))

                             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
o6 = map(ringP8,ringP8,{- 43x  - 58x x  + 98x  - 163x x  + 110x x  - 158x  + 158x x  - 131x x  + 88x x  + 57x  + 68x x  + 35x x  + 21x x  - 8x x  + 116x  - 67x x  + 121x x  - 23x x  - 119x x  + 153x x  - 51x  - 26x x  + 71x x  - 57x x  + 87x x  + 61x x  - 101x x  + 61x  - 151x x  - 121x x  + 28x x  + 132x x  - 78x x  - 78x x  - 157x x  + 55x  - 115x x  - 37x x  + 95x x  + 116x x  - 60x x  - 81x x  - 123x x  - 23x x  + 156x , - 120x  + 111x x  + 117x  + 151x x  - 58x x  + 96x  - 86x x  + 136x x  + 111x x  + 109x  - 20x x  + 109x x  + 122x x  - 34x x  + 35x  + 133x x  + 160x x  - 143x x  + 116x x  + 113x x  + 129x  - 126x x  + 112x x  - 86x x  + 33x x  + 19x x  - 5x x  + 107x  + 32x x  + 128x x  - 78x x  - 134x x  - 113x x  - 49x x  - 102x x  + 9x  + 98x x  + 58x x  - 148x x  + 113x x  - 44x x  + 107x x  + x x  - 6x x  - 109x , 74x  - 8x x  + 33x  - 96x x  - 128x x  + 41x  + 79x x  + 138x x  - 141x x  + 29x  + 137x x  - 122x x  + 114x x  + 44x x  - 141x  - 25x x  + 115x x  - 164x x  + 22x x  - 148x x  - 58x  + 31x x  + 145x x  - 117x x  + 133x x  + 121x x  - 21x x  + 88x  + 135x x  + 97x x  - 132x x  + 48x x  + 34x x  - 74x x  + 129x x  + 61x  - 117x x  + 20x x  - 147x x  + 66x x  + 43x x  + 46x x  - 61x x  + 134x x  + 17x , - 47x  - 116x x  - 73x  - 50x x  - 120x x  - 26x  - 135x x  - 39x x  + 155x x  + 162x  - 134x x  - 102x x  + 86x x  + 66x x  - 69x  + 131x x  + 74x x  + 6x x  + 72x x  - 120x x  - 23x  - 42x x  + 39x x  - 3x x  + 125x x  + 37x x  - 89x x  + 62x  - 77x x  + 163x x  - 134x x  - 135x x  - 107x x  + 109x x  + 27x x  + 138x  - 67x x  + 89x x  - 32x x  - 26x x  + 48x x  + 13x x  - 118x x  + 100x x  + 74x , - 75x  - x x  + 68x  - 153x x  - 2x x  - 79x  + 148x x  - 87x x  + 61x x  + 129x  - 38x x  + 58x x  - 127x x  + 91x x  - 32x  - 61x x  + 126x x  - 10x x  - 63x x  - 39x x  + 163x  + 96x x  - 20x x  + 94x x  + 136x x  + 86x x  + 111x x  - 35x  + 163x x  - 22x x  - 49x x  - 93x x  + 34x x  - 135x x  - 62x x  - 101x  + 151x x  + 21x x  - 79x x  - 22x x  - 77x x  - 137x x  - 92x x  + 47x x  + 113x , - 96x  + 161x x  - 118x  - 25x x  - 144x x  - 125x  - 107x x  + 98x x  - 28x x  + 104x  + 46x x  + 18x x  + 11x x  - 65x x  - 98x  + 163x x  - 26x x  - 140x x  + 90x x  + 60x x  - 63x  + 41x x  - 136x x  - 61x x  + 133x x  - 157x x  - 22x x  - 29x  + 161x x  - 56x x  + 148x x  - 30x x  + 79x x  - 116x x  - 63x x  - 140x  - 36x x  + 141x x  + 155x x  - 46x x  + 9x x  - 16x x  - 36x x  + 85x x  - 77x , - 146x  + 6x x  + 79x  - 162x x  - 104x x  + 29x  + 66x x  - 86x x  - 66x x  + 133x  + 8x x  - 84x x  - 134x x  - 107x x  + 74x  + 94x x  + 89x x  + 124x x  - 88x x  - 108x x  - 54x  - 80x x  - 150x x  - 47x x  - 22x x  + 104x x  - 50x x  + 164x  + 11x x  - 113x x  + 74x x  + 43x x  - 75x x  - 84x x  - 25x x  - 67x  + 122x x  - 13x x  - 29x x  - 124x x  + 53x x  + 62x x  + 101x x  - 58x x  - 91x , 81x  + 141x x  - 62x  - 108x x  - 30x x  + 93x  + 86x x  - 160x x  + 50x x  + 148x  - 138x x  + 124x x  - 162x x  + 81x x  - 149x  + 134x x  + 20x x  - 71x x  + 121x x  + 126x x  - 47x  + 104x x  - 37x x  + 146x x  - 111x x  - 161x x  + 79x x  - 109x  - 21x x  + 58x x  - 133x x  + 54x x  - 125x x  + 14x x  - 25x x  + 91x  - 4x x  - 98x x  - 134x x  - 56x x  + 152x x  + 41x x  + 109x x  - 73x x  + 141x , 12x  - 91x x  + 107x  - 85x x  + 58x x  - 103x  + 28x x  + 165x x  + 71x x  - 164x  + 12x x  - 83x x  - 12x x  + 40x x  + 122x  + 43x x  + 115x x  + 93x x  + 39x x  - 49x x  + 142x  + 165x x  - 84x x  + 76x x  - 18x x  - 87x x  - 52x x  + 38x  + 46x x  + 131x x  - 54x x  + 95x x  + 163x x  + 149x x  - 87x x  + 109x  - 104x x  + 137x x  - 50x x  + 31x x  - 50x x  - 110x x  + 152x x  + 28x x  - 149x })
                             0      0 1      1       0 2       1 2       2       0 3       1 3      2 3      3      0 4      1 4      2 4     3 4       4      0 5       1 5      2 5       3 5       4 5      5      0 6      1 6      2 6      3 6      4 6       5 6      6       0 7       1 7      2 7       3 7      4 7      5 7       6 7      7       0 8      1 8      2 8       3 8      4 8      5 8       6 8      7 8       8        0       0 1       1       0 2      1 2      2      0 3       1 3       2 3       3      0 4       1 4       2 4      3 4      4       0 5       1 5       2 5       3 5       4 5       5       0 6       1 6      2 6      3 6      4 6     5 6       6      0 7       1 7      2 7       3 7       4 7      5 7       6 7     7      0 8      1 8       2 8       3 8      4 8       5 8    6 8     7 8       8     0     0 1      1      0 2       1 2      2      0 3       1 3       2 3      3       0 4       1 4       2 4      3 4       4      0 5       1 5       2 5      3 5       4 5      5      0 6       1 6       2 6       3 6       4 6      5 6      6       0 7      1 7       2 7      3 7      4 7      5 7       6 7      7       0 8      1 8       2 8      3 8      4 8      5 8      6 8       7 8      8       0       0 1      1      0 2       1 2      2       0 3      1 3       2 3       3       0 4       1 4      2 4      3 4      4       0 5      1 5     2 5      3 5       4 5      5      0 6      1 6     2 6       3 6      4 6      5 6      6      0 7       1 7       2 7       3 7       4 7       5 7      6 7       7      0 8      1 8      2 8      3 8      4 8      5 8       6 8       7 8      8       0    0 1      1       0 2     1 2      2       0 3      1 3      2 3       3      0 4      1 4       2 4      3 4      4      0 5       1 5      2 5      3 5      4 5       5      0 6      1 6      2 6       3 6      4 6       5 6      6       0 7      1 7      2 7      3 7      4 7       5 7      6 7       7       0 8      1 8      2 8      3 8      4 8       5 8      6 8      7 8       8       0       0 1       1      0 2       1 2       2       0 3      1 3      2 3       3      0 4      1 4      2 4      3 4      4       0 5      1 5       2 5      3 5      4 5      5      0 6       1 6      2 6       3 6       4 6      5 6      6       0 7      1 7       2 7      3 7      4 7       5 7      6 7       7      0 8       1 8       2 8      3 8     4 8      5 8      6 8      7 8      8        0     0 1      1       0 2       1 2      2      0 3      1 3      2 3       3     0 4      1 4       2 4       3 4      4      0 5      1 5       2 5      3 5       4 5      5      0 6       1 6      2 6      3 6       4 6      5 6       6      0 7       1 7      2 7      3 7      4 7      5 7      6 7      7       0 8      1 8      2 8       3 8      4 8      5 8       6 8      7 8      8     0       0 1      1       0 2      1 2      2      0 3       1 3      2 3       3       0 4       1 4       2 4      3 4       4       0 5      1 5      2 5       3 5       4 5      5       0 6      1 6       2 6       3 6       4 6      5 6       6      0 7      1 7       2 7      3 7       4 7      5 7      6 7      7     0 8      1 8       2 8      3 8       4 8      5 8       6 8      7 8       8     0      0 1       1      0 2      1 2       2      0 3       1 3      2 3       3      0 4      1 4      2 4      3 4       4      0 5       1 5      2 5      3 5      4 5       5       0 6      1 6      2 6      3 6      4 6      5 6      6      0 7       1 7      2 7      3 7       4 7       5 7      6 7       7       0 8       1 8      2 8      3 8      4 8       5 8       6 8      7 8       8

o6 : RingMap ringP8 <--- ringP8
i7 : time degreeOfRationalMap phi'
     -- used 8.99701 seconds

o7 = 14

Caveat

This method can be generalized for the case when the source of the rational map is just parameterized by a projective space.

See also

Ways to use degreeOfRationalMap :