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inverse(RationalMap) -- inverse of a birational map

Description

inverse(phi,Certify=>true) is the same as inverseMap(phi,Certify=>true,Verbose=>false), while inverse(phi,Certify=>false) applies a middle ground approach between inverseMap(phi,Certify=>true) and inverseMap(phi,Certify=>false). The procedure for the latter is as follows: It first computes the inverse map of phi using psi = inverseMap phi. Then it is checked that psi phi p == p and phi psi q == q, where p,q are, respectively, a random point on the source and the target of phi. Finally, if the tests pass, the command forceInverseMap(phi,psi) is invoked and psi is returned.

i1 : R = QQ[x_0..x_4]; phi = rationalMap minors(4,random(R^{4:1},R^5)) -- special Cremona transformation of P^4 of type (4,4)

o2 = -- rational map --
     source: Proj(QQ[x , x , x , x , x ])
                      0   1   2   3   4
     target: Proj(QQ[x , x , x , x , x ])
                      0   1   2   3   4
     defining forms: {
                      149869 4   782490613 3     18482371601 2 2   819073079   3   187789 4   1364748967 3     661142958937 2       295309047329   2     24633811 3     80609387 2 2   627335093171     2   64111908347 2 2   10742497   3   1038063727   3   463864 4   23348164619 3     308230651567 2       132496223449   2     746419537 3     18303121889 2       252607515691           23524322789 2       23898152759   2     869384658167   2     2083490191 3     18332039953 2 2   15125040407     2   8858126881 2 2   16108173989     2   6419246567     2   2189106919 2 2   272275693   3   51200969   3   8012407   3   236333 4   913877173 3     3205319143 2       27115564519   2     411279151 3     83231814691 2       73963606468403           246506069 2       480723995903   2     141607467209   2     713958589 3     61868232101 2       71171313775487           638186063057 2       2176335571321           21417746291           3843027023 2       56840321383   2     17888753819   2     900109855   2     40822381 3     16643505173 2 2   2934917471077     2   17020063939 2 2   23736687907     2   50467602883     2   20389552243 2 2   304387378639     2   236593487273     2   33279595319     2   10982102863 2 2   1560434321   3   4669122967   3   1805282077   3   6465149767   3   120559 4
                      ------x  + ---------x x  - -----------x x  - ---------x x  + ------x  + ----------x x  + ------------x x x  - ------------x x x  + --------x x  + --------x x  + ------------x x x  - -----------x x  + --------x x  - ----------x x  + ------x  - -----------x x  - ------------x x x  - ------------x x x  + ---------x x  + -----------x x x  - ------------x x x x  + -----------x x x  + -----------x x x  - ------------x x x  - ----------x x  + -----------x x  - -----------x x x  + ----------x x  + -----------x x x  + ----------x x x  - ----------x x  + ---------x x  + --------x x  - -------x x  - ------x  - ---------x x  - ----------x x x  - -----------x x x  - ---------x x  - -----------x x x  - --------------x x x x  - ---------x x x  - ------------x x x  - ------------x x x  - ---------x x  + -----------x x x  - --------------x x x x  + ------------x x x  + -------------x x x x  - -----------x x x x  - ----------x x x  + -----------x x x  + -----------x x x  + ---------x x x  - --------x x  - -----------x x  - -------------x x x  - -----------x x  - -----------x x x  - -----------x x x  - -----------x x  + ------------x x x  - ------------x x x  + -----------x x x  + -----------x x  - ----------x x  - ----------x x  - ----------x x  + ----------x x  - ------x ,
                       20160 0    16934400 0 1    114307200  0 1    65318400 0 1    21600 1    67737600  0 2    1016064000  0 1 2    3429216000  0 1 2    544320  1 2    5080320 0 2    1714608000  0 1 2    285768000  1 2    1587600 0 2    11430720  1 2    59535 2    148176000  0 3     592704000  0 1 3     762048000  0 1 3    10206000 1 3     71124480  0 2 3    5334336000  0 1 2 3     68584320  1 2 3     88905600  0 2 3    2400451200  1 2 3    20003760  2 3     98784000  0 3     76204800  0 1 3    81648000  1 3     29635200  0 2 3    21168000  1 2 3     9525600  2 3    3528000  0 3    1944000 1 3    226800 2 3    9450  3   169344000 0 4    57153600  0 1 4    914457600  0 1 4    13608000 1 4    8001504000 0 2 4    144027072000  0 1 2 4    3402000  1 2 4    4800902400  0 2 4     889056000  1 2 4    12348000 2 4    5334336000 0 3 4     80015040000  0 1 3 4    3674160000  1 3 4     8001504000  0 2 3 4    2000376000 1 2 3 4    74088000  2 3 4     83349000  0 3 4     69984000  1 3 4    8001504  2 3 4    388800  3 4    4572288000 0 4    68584320000  0 1 4    408240000  1 4    1125211500 0 2 4    381024000  1 2 4    666792000  2 4    60011280000 0 3 4    2057529600  1 3 4    937676250  2 3 4     85730400  3 4   1428840000 0 4    408240000 1 4    357210000 2 4   4286520000 3 4   486000 4
                      
                        420641 4   2786860733 3     1195078457 2 2   771090851   3   6403999 4   6349833997 3     14770761451 2       809959481   2     854002123 3     1194212407 2 2   548833195261     2   9969298253 2 2   124512641   3   13015502747   3   12168557 4   2760332797 3     19818912653 2       260633491   2     6985945457 3     39778976537 2       2393249221327           2084699540143 2       3119038961   2     762553261009   2     2392520731 3     39219663199 2 2   333328105753     2   313490299 2 2   46174554563     2   176957615459     2   10904423987 2 2   2184395041   3   109856927   3   5213790311   3   3203839 4   349286107 3     581957237 2       113939091347   2     186071591 3     1896897882827 2       814934281607           27790583 2       307964189   2     390203309459   2     12813855167 3     1700452301 2       71820937297           2232985325569 2       4761866701987           16030202072981           1272542857301 2       11598010594   2     2643230995   2     34350673633   2     154904293 3     1334057353 2 2   6979929053     2   5585595193 2 2   67272005429     2   642473361133     2   3524615371 2 2   66536189773     2   1618288961     2   1502291957     2   675621047 2 2   67119929   3   30046843   3   876948217   3   30351707   3   2887 4
                      - ------x  + ----------x x  - ----------x x  + ---------x x  + -------x  - ----------x x  + -----------x x x  - ---------x x x  + ---------x x  - ----------x x  - ------------x x x  - ----------x x  + ---------x x  - -----------x x  + --------x  + ----------x x  - -----------x x x  + ---------x x x  + ----------x x  + -----------x x x  - -------------x x x x  + -------------x x x  - ----------x x x  - ------------x x x  - ----------x x  + -----------x x  - ------------x x x  + ---------x x  - -----------x x x  + ------------x x x  - -----------x x  - ----------x x  + ---------x x  + ----------x x  - -------x  + ---------x x  - ---------x x x  - ------------x x x  - ---------x x  + -------------x x x  + ------------x x x x  + --------x x x  + ---------x x x  + ------------x x x  + -----------x x  + ----------x x x  - -----------x x x x  + -------------x x x  + -------------x x x x  + --------------x x x x  - -------------x x x  - -----------x x x  + ----------x x x  - -----------x x x  - ---------x x  - ----------x x  - ----------x x x  - ----------x x  + -----------x x x  - ------------x x x  - ----------x x  - -----------x x x  - ----------x x x  + ----------x x x  + ---------x x  + --------x x  - --------x x  - ---------x x  - --------x x  - ----x ,
                         28224 0    63504000  0 1    31752000  0 1    27216000 0 1    324000 1    76204800  0 2    342921600  0 1 2    10584000 0 1 2    7776000  1 2    12700800  0 2    2286144000  0 1 2    23328000  1 2    10886400 0 2     97977600  1 2    272160  2    22226400  0 3    333396000  0 1 3    52488000 0 1 3    40824000  1 3    250047000  0 2 3    36006768000  0 1 2 3     2571912000  1 2 3    12502350  0 2 3    2057529600  1 2 3    11430720  2 3    2333772000 0 3    1000188000  0 1 3    1275750  1 3    145860750  0 2 3     222264000  1 2 3     26671680  2 3    11907000  0 3    2646000  1 3    33339600  2 3    132300 3    19051200 0 4    9072000  0 1 4    2857680000  0 1 4    3240000  1 4    12002256000  0 2 4    2667168000  0 1 2 4    5832000 1 2 4    3402000  0 2 4    3429216000  1 2 4    857304000  2 4    62511750  0 3 4    262440000  0 1 3 4     4286520000  1 3 4    50409475200  0 2 3 4     36006768000  1 2 3 4    18003384000  2 3 4     24310125  0 3 4     8001504  1 3 4    700131600  2 3 4    16669800 3 4    190512000 0 4   2857680000 0 1 4    68040000  1 4    1600300800 0 2 4    5715360000  1 2 4    214326000 2 4    2500470000 0 3 4    25515000  1 3 4    27783000  2 3 4    5556600  3 4   47628000 0 4    1360800 1 4   142884000 2 4   13395375 3 4   6804 4
                      
                        561079 4   5003323307 3     97427345327 2 2   1467949949   3   86977 4   10508353 3     36142167529 2       270387777431   2     656224057 3     6003594533 2 2   7705482749     2   113341477 2 2   29877391   3   15553163   3   2318507 4   32754865939 3     1024224828667 2       9222397669   2     265032973 3     9504423163 2       36636577774711           207680427287 2       1266256929383   2     4685270777   2     7693501 3     9946991591 2 2   16142827037     2   4465470581 2 2   41663119759     2   69433172987     2   29174750881 2 2   8025079147   3   1880953   3   7212009017   3   39169 4   169129183 3     942658275907 2       17212691389   2     34522259 3     183905429311 2       178717759547           62556525347 2       15495920291   2     35877695837   2     728430041 3     525707858609 2       230817747751           92926962077 2       3881435678233           30952544921           544612570297 2       14896627313   2     19179845011   2     448988849929   2     81256486 3     997283467 2 2   216563459297     2   653724407 2 2    8638051     2   28552340729     2   1503574259 2 2   14542895587     2   1524947411     2   111685523497     2   5760874381 2 2   6996200033   3   1119325853   3   1242610319   3   36944143   3   103651 4
                      - ------x  + ----------x x  - -----------x x  + ----------x x  + -----x  + --------x x  - -----------x x x  - ------------x x x  + ---------x x  + ----------x x  - ----------x x x  + ---------x x  + --------x x  + --------x x  - -------x  + -----------x x  - -------------x x x  + ----------x x x  + ---------x x  + ----------x x x  - --------------x x x x  + ------------x x x  + -------------x x x  + ----------x x x  - -------x x  - ----------x x  - -----------x x x  + ----------x x  - -----------x x x  + -----------x x x  - -----------x x  - ----------x x  + -------x x  + ----------x x  + -----x  + ---------x x  + ------------x x x  + -----------x x x  + --------x x  - ------------x x x  + ------------x x x x  + -----------x x x  - -----------x x x  + -----------x x x  - ---------x x  - ------------x x x  + ------------x x x x  + -----------x x x  - -------------x x x x  + -----------x x x x  - ------------x x x  - -----------x x x  + -----------x x x  - ------------x x x  + --------x x  + ---------x x  + ------------x x x  + ---------x x  - --------x x x  + -----------x x x  + ----------x x  - -----------x x x  + ----------x x x  + ------------x x x  + ----------x x  + ----------x x  + ----------x x  + ----------x x  - --------x x  + ------x ,
                         56448 0    19051200  0 1    190512000  0 1    13608000  0 1    4050 1    714420  0 2    685843200  0 1 2     381024000  0 1 2    6804000  1 2    91445760  0 2    42865200  0 1 2    2041200  1 2    653184  0 2    6123600 1 2    816480 2     44452800  0 3     666792000   0 1 3    128595600 0 1 3    2551500  1 3     9878400  0 2 3     18003384000  0 1 2 3     367416000  1 2 3     4800902400  0 2 3    257191200 1 2 3    107163 2 3    13335840  0 3     47628000  0 1 3    35721000  1 3     34574400  0 2 3    100018800  1 2 3    133358400  2 3    33339600  0 3    37800  1 3    33339600  2 3    3780 3    7144200  0 4    5143824000  0 1 4     7620480   0 1 4    162000  1 4    1028764800  0 2 4    1469664000  0 1 2 4     81648000  1 2 4     82301184  0 2 4    190512000  1 2 4    14288400 2 4    1000188000  0 3 4     300056400  0 1 3 4    428652000  1 3 4     6001128000  0 2 3 4     55566000  1 2 3 4    6001128000  2 3 4     16669800  0 3 4    1000188000 1 3 4    3500658000  2 3 4    2083725 3 4    31752000 0 4     428652000  0 1 4    2916000  1 4   69984000 0 2 4    214326000  1 2 4    95256000  2 4     62511750  0 3 4    26790750  1 3 4    4000752000  2 3 4    125023500 3 4    857304000 0 4    20412000  1 4    114307200 2 4   14288400 3 4    97200 4
                      
                        815257 4   7060569421 3     19831003091 2 2   5436785981   3   7761221 4   16909987433 3     240482195549 2       35799188651   2     617694809 3     165186711677 2 2   28474654221911     2   298501773901 2 2   4684848671   3   655371181   3   5638165 4   27892618183 3     224732253439 2       3927697425241   2     5381747461 3     276532115 2       3719655387523           690511871513 2       726522786541   2     348836298901   2     24284843 3     32247662281 2 2   578849997643     2   617677997 2 2   279787636631     2   38102737621     2   8389162319 2 2   2243919511   3   345434819   3   169054597   3   380351 4   558495113 3     22034381189 2       613583636879   2     32310643 3     9383284012489 2       709482289183           130034281117 2       939839793103   2     83180063467   2     8831673817 3     340210568531 2       6151870667597           246582589 2       1006476993247           8443189256947           3193698546041 2       16646877628   2     9254280013   2     13926393011   2     15281741 3     1070357189 2 2   356382221     2   8529723001 2 2   572680113539     2   289106338939     2   1164797059 2 2   43722061379     2   51708533329     2   3209584919     2   47640239 2 2   7057559869   3   1288452083   3   332861881   3   42542393   3    237973 4
                      - ------x  + ----------x x  - -----------x x  - ----------x x  - -------x  - -----------x x  + ------------x x x  + -----------x x x  - ---------x x  - ------------x x  + --------------x x x  + ------------x x  - ----------x x  + ---------x x  - -------x  + -----------x x  - ------------x x x  - -------------x x x  - ----------x x  + ---------x x x  + -------------x x x x  - ------------x x x  + ------------x x x  + ------------x x x  + --------x x  + -----------x x  - ------------x x x  + ---------x x  + ------------x x x  - -----------x x x  + ----------x x  - ----------x x  + ---------x x  - ---------x x  + ------x  + ---------x x  + -----------x x x  + ------------x x x  + --------x x  + -------------x x x  + ------------x x x x  + ------------x x x  - ------------x x x  + -----------x x x  - ----------x x  - ------------x x x  - -------------x x x x  - ---------x x x  + -------------x x x x  - -------------x x x x  + -------------x x x  - -----------x x x  - ----------x x x  - -----------x x x  - --------x x  + ----------x x  + ---------x x x  + ----------x x  + ------------x x x  + ------------x x x  + ----------x x  - -----------x x x  - -----------x x x  - ----------x x x  - --------x x  + ----------x x  + ----------x x  + ---------x x  - --------x x  + -------x ,
                        169344 0    54432000  0 1    142884000  0 1    18144000  0 1    324000 1    457228800  0 2     419904000  0 1 2     47628000  0 1 2    54432000 1 2    1828915200  0 2     82301184000  0 1 2    1143072000  1 2    57153600  0 2    8467200  1 2    254016 2    177811200  0 3     333396000  0 1 3     2057529600  0 1 3    40824000  1 3     333396  0 2 3    24004512000  0 1 2 3    2571912000  1 2 3    1066867200  0 2 3    1371686400  1 2 3    238140  2 3     88905600  0 3     244944000  0 1 3    40824000 1 3     400075200  0 2 3    171460800  1 2 3    57153600  2 3     3175200  0 3    1360800  1 3    1587600  2 3    3780  3    45722880 0 4    1143072000 0 1 4    1428840000  0 1 4    6480000 1 4    288054144000 0 2 4    5334336000  0 1 2 4     653184000  1 2 4   144027072000 0 2 4    2286144000 1 2 4    666792000 2 4    3000564000  0 3 4    10287648000  0 1 3 4    1693440  1 3 4     8001504000  0 2 3 4    144027072000 1 2 3 4    84015792000  2 3 4     93767625  0 3 4    71442000  1 3 4    480090240  2 3 4    238140  3 4    228614400 0 4    3402000  0 1 4    272160000 1 4    32006016000 0 2 4    22861440000 1 2 4    571536000 2 4    600112800  0 3 4    571536000  1 3 4   2400451200 2 3 4    9525600 3 4   2857680000 0 4    136080000 1 4   190512000 2 4    6804000 3 4   1134000 4
                      
                        9484513 4   203474903 3     146682548251 2 2   5305140497   3   79727 4   46999672073 3     9854555140589 2       3661901136779   2     69292243 3     1106213651 2 2   2548967007707     2   3504318061 2 2   15171468329   3   2620882819   3   397099 4   10962737447 3     3859607251 2       50351276201   2     3791471 3     1154844662123 2       391933798499           46441792423 2       32318066003   2     395696461301   2     6595596629 3     46015608493 2 2   107076650353     2   604631497 2 2   239074683137     2   27185887429     2   14221301833 2 2   5824914829   3   1668949   3   82767059   3   172324 4   149669483 3     497577251063 2       2552241433481   2     1454072113 3     16348418701907 2       8728116619543           676570135231 2       7596809117701   2     186756875951   2     9666609499 3     11907700169 2       7489176428071           1050244487 2       12265987537187           7121627928481           2202470968921 2       246142289899   2     179061497   2     179297416343   2     380172803 3     100420273 2 2   22091593703     2   3080464099 2 2   21741199903     2   41955024017     2   4329734251 2 2   38574397831     2   16314237601     2   109944994619     2   1670238943 2 2   103111517   3   576721   3    7327739   3   1288046897   3    2818 4
                      - -------x  + ---------x x  + ------------x x  - ----------x x  - -----x  - -----------x x  + -------------x x x  + -------------x x x  - --------x x  - ----------x x  + -------------x x x  - ----------x x  - -----------x x  + ----------x x  + ------x  + -----------x x  + ----------x x x  - -----------x x x  - -------x x  + -------------x x x  + ------------x x x x  - -----------x x x  + -----------x x x  - ------------x x x  + ----------x x  - -----------x x  - ------------x x x  - ---------x x  - ------------x x x  - -----------x x x  + -----------x x  - ----------x x  + -------x x  + --------x x  + ------x  + ---------x x  - ------------x x x  - -------------x x x  - ----------x x  + --------------x x x  + -------------x x x x  - ------------x x x  + -------------x x x  + ------------x x x  + ----------x x  + -----------x x x  - -------------x x x x  - ----------x x x  + --------------x x x x  - -------------x x x x  + -------------x x x  - ------------x x x  + ---------x x x  - ------------x x x  + ---------x x  - ---------x x  - -----------x x x  - ----------x x  + -----------x x x  + -----------x x x  + ----------x x  - -----------x x x  - -----------x x x  - ------------x x x  - ----------x x  + ---------x x  - ------x x  - --------x x  - ----------x x  - -----x
                         290304 0    3888000  0 1    2939328000  0 1    163296000 0 1   20250 1    228614400  0 2    41150592000  0 1 2    41150592000  0 1 2    3888000 1 2     3572100  0 2    10287648000  0 1 2    342921600 1 2    114307200  0 2    63504000  1 2    25200 2     76204800  0 3    42336000  0 1 3    428652000  0 1 3    212625 1 3     5334336000  0 2 3    9601804800  0 1 2 3    489888000  1 2 3    222264000  0 2 3    12002256000 1 2 3    66679200  2 3    666792000  0 3     666792000  0 1 3    47628000 1 3    1333584000  0 2 3    444528000  1 2 3    777924000  2 3    55566000  0 3    105840 1 3    3472875 2 3    11025 3    4665600  0 4    2939328000  0 1 4     4898880000  0 1 4    29160000  1 4     41150592000  0 2 4    20575296000  0 1 2 4    4898880000  1 2 4    20575296000  0 2 4    1371686400  1 2 4    95256000  2 4     40824000  0 3 4     8573040000  0 1 3 4    11664000  1 3 4     24004512000  0 2 3 4    34292160000  1 2 3 4    12002256000  2 3 4     333396000  0 3 4    5292000  1 3 4    1333584000  2 3 4    3969000  3 4    6804000  0 4    272160000  0 1 4    58320000  1 4    190512000  0 2 4    4898880000 1 2 4    190512000 2 4    476280000  0 3 4    204120000  1 3 4    2857680000  2 3 4    23814000  3 4    30618000 0 4    46656 1 4   12757500 2 4    51030000  3 4   30375 4
                     }

o2 : RationalMap (rational map from PP^4 to PP^4)
 
i3 : time inverse phi
 -- used 0.0767488s (cpu); 0.0772953s (thread); 0s (gc)

o3 = -- rational map --
     source: Proj(QQ[x , x , x , x , x ])
                      0   1   2   3   4
     target: Proj(QQ[x , x , x , x , x ])
                      0   1   2   3   4
     defining forms: {
                                         4                    3                      2 2                     3                  4                    3                      2                          2                     3                      2 2                        2                   2 2                      3                     3                  4                   3                      2                          2                      3                       2                                                    2                         2                        2                     3                      2 2                        2                    2 2                        2                        2                    2 2                      3                      3                    3                   4                     3                       2                          2                     3                     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                       3                      3                       3                    4
                      - 3091483845273600x  + 3852426902768640x x  - 2322800320075200x x  + 602077207475520x x  - 32377854781440x  + 4408967074982400x x  - 7284481307046720x x x  + 3730110554983260x x x  - 455809961350800x x  + 1155391697466720x x  - 1467032075626572x x x  + 439237953929880x x  - 1399143158148960x x  + 153655904472840x x  - 56964156137280x  - 412328359457280x x  + 2676950653043520x x x  + 1877987324848320x x x  - 1338568632737280x x  - 15381011929752000x x x  + 12741870868444536x x x x  - 4115214528674256x x x  - 395574627427056x x x  + 3639366604555920x x x  - 636226579394784x x  + 8875863133804800x x  + 2251556907526080x x x  - 3137853981561600x x  + 5332040430581088x x x  - 5076424885107072x x x  + 5403047891774400x x  + 3088313127809280x x  - 1986115710539520x x  - 91363834420992x x  - 167661689287680x  - 14168140126963200x x  + 14318698812519360x x x  - 2432577668070480x x x  + 121720688712000x x  + 974653908735360x x x  - 11263442842340360x x x x  + 4643500060853640x x x  - 1610446745169360x x x  - 2517484698314880x x x  - 67422007896768x x  - 3054218285932800x x x  + 32772309124907520x x x x  - 18063023556594240x x x  - 37980241544906080x x x x  + 35900192769036960x x x x  - 7148595213201120x x x  + 57612199799521920x x x  - 36247472684442880x x x  + 29617398392382080x x x  - 2525608367546880x x  - 1408372992902400x x  - 8997447126539200x x x  + 6705417490178400x x  + 6482977849988640x x x  - 13350559981851120x x x  - 291084893838480x x  - 14544977281022720x x x  + 30729035211721280x x x  - 21337690456106080x x x  + 36164909117299200x x  + 9530061023347200x x  - 11888845281264000x x  + 2409695178634800x x  - 11010681397968000x x  + 2616835749192000x ,
                                         0                    0 1                    0 1                   0 1                  1                    0 2                    0 1 2                    0 1 2                   1 2                    0 2                    0 1 2                   1 2                    0 2                   1 2                  2                   0 3                    0 1 3                    0 1 3                    1 3                     0 2 3                     0 1 2 3                    1 2 3                   0 2 3                    1 2 3                   2 3                    0 3                    0 1 3                    1 3                    0 2 3                    1 2 3                    2 3                    0 3                    1 3                  2 3                   3                     0 4                     0 1 4                    0 1 4                   1 4                   0 2 4                     0 1 2 4                    1 2 4                    0 2 4                    1 2 4                  2 4                    0 3 4                     0 1 3 4                     1 3 4                     0 2 3 4                     1 2 3 4                    2 3 4                     0 3 4                     1 3 4                     2 3 4                    3 4                    0 4                    0 1 4                    1 4                    0 2 4                     1 2 4                   2 4                     0 3 4                     1 3 4                     2 3 4                     3 4                    0 4                     1 4                    2 4                     3 4                    4
                      
                                       4                     3                       2 2                      3                    4                    3                       2                           2                      3                      2 2                        2                   2 2                     3                      3                   4                    3                       2                           2                       3                       2                                                  2                          2                         2                     3                       2 2                        2                    2 2                         2                        2                   2 2                       3                      3                      3                  4                    3                       2                           2                      3                      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                      3                      3                      3                    4
                      3429282044160000x  - 12591283668768000x x  + 17434712390961600x x  - 8361743515738200x x  + 1225455909073200x  + 4151415342816000x x  - 18620455397987520x x x  + 15887368367436300x x x  - 3696272364884400x x  + 2360959219936800x x  - 2128572984953940x x x  - 805482682721640x x  + 361614797318880x x  + 2930780202854616x x  - 137585103042240x  - 1182542445504000x x  + 30220580131378560x x x  - 42550174412759160x x x  + 12552065714462400x x  + 10845047493302400x x x  + 7375162689510000x x x x  + 592689733861680x x x  + 4231408805956080x x x  - 13487883897203856x x x  + 314997040118880x x  - 23294434073299200x x  - 1922596640828640x x x  + 9515971456629120x x  - 25144808938583040x x x  + 1410969367916640x x x  - 434427250235328x x  + 14818874047729920x x  - 2571958272260160x x  + 2600418840969600x x  + 15748075848960x  + 8458549783488000x x  - 20273200713964800x x x  + 14846902757160600x x x  - 3468046029890400x x  + 9157976827208640x x x  - 21298818455693520x x x x  + 8477794172829240x x x  + 1765529659002960x x x  + 2298040164927360x x x  - 1213350879599712x x  - 22874713561029120x x x  + 68204973199923840x x x x  - 36213139365031440x x x  + 2295409634683040x x x x  + 6483539890684080x x x x  + 7203066054730560x x x  - 13432970137507200x x x  - 15319605490418880x x x  - 12986289190224000x x x  + 14059362367330560x x  + 4531134067219200x x  - 11376159405554400x x x  + 6455295973501200x x  + 7051800456008960x x x  - 8388716004254880x x x  - 163672218123120x x  - 21775695483140480x x x  + 35552216652472800x x x  - 2506883269907520x x x  + 414478571451840x x  + 3241693646060800x x  - 5979316392957600x x  + 3000330654375600x x  - 9888313751769600x x  + 1784976249672000x ,
                                       0                     0 1                     0 1                    0 1                    1                    0 2                     0 1 2                     0 1 2                    1 2                    0 2                    0 1 2                   1 2                   0 2                    1 2                   2                    0 3                     0 1 3                     0 1 3                     1 3                     0 2 3                    0 1 2 3                   1 2 3                    0 2 3                     1 2 3                   2 3                     0 3                    0 1 3                    1 3                     0 2 3                    1 2 3                   2 3                     0 3                    1 3                    2 3                  3                    0 4                     0 1 4                     0 1 4                    1 4                    0 2 4                     0 1 2 4                    1 2 4                    0 2 4                    1 2 4                    2 4                     0 3 4                     0 1 3 4                     1 3 4                    0 2 3 4                    1 2 3 4                    2 3 4                     0 3 4                     1 3 4                     2 3 4                     3 4                    0 4                     0 1 4                    1 4                    0 2 4                    1 2 4                   2 4                     0 3 4                     1 3 4                    2 3 4                   3 4                    0 4                    1 4                    2 4                    3 4                    4
                      
                                       4                    3                      2 2                     3                   4                    3                       2                          2                     3                      2 2                        2                    2 2                     3                      3                   4                  3                       2                          2                      3                       2                                                    2                          2                         2                    3                      2 2                        2                    2 2                        2                        2                    2 2                      3                     3                     3                 4                    3                       2                           2                      3                       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                       3                       3                       3                    4
                      2223334987776000x  + 2078831578199040x x  - 1315093204454400x x  - 332656127102880x x  + 166390817816160x  - 6652790231270400x x  + 13156877381316480x x x  - 6894650674626960x x x  + 559070350191990x x  - 5119543510623360x x  + 9401478986517408x x x  - 3793629683200140x x  - 895218819027120x x  + 3273500393168670x x  - 165638167455600x  + 61155740620800x x  - 18506916715633920x x x  + 2890150995745440x x x  + 2551657745767200x x  + 15557871465158400x x x  - 11751937140509664x x x x  + 7228028698704324x x x  + 2602532644981920x x x  - 12349382657318700x x x  + 86214660812280x x  + 2069163786816000x x  - 7443019944239040x x x  + 3159660819641760x x  - 1769119707015360x x x  + 1767436287089256x x x  + 1562820193354896x x  - 2243703838199040x x  + 822287731688640x x  - 123156222668640x x  - 8034260209920x  + 4410398084843520x x  + 26698085414972160x x x  - 32895183229396320x x x  + 6726073928537880x x  - 11252913772154880x x x  + 66150026437236560x x x x  - 29320853020661790x x x  - 8067188401500816x x x  + 24355994670064890x x x  - 2323426768213320x x  + 12439437285488640x x x  - 133929453403441920x x x x  + 79255963348490880x x x  + 16455137472950848x x x x  - 78817617234899928x x x x  + 6367305862699992x x x  + 778437911462400x x x  + 7417955776230720x x x  + 5329836296705088x x x  - 2590571251578240x x  - 22006456622008320x x  + 46777577500615360x x x  - 18641974584756600x x  - 32739348545467200x x x  + 49469589965617920x x x  - 10489133281795560x x  + 77538073878836480x x x  - 126776856246636960x x x  + 35480356251238080x x x  - 9546660540144000x x  - 16210477172736000x x  + 21133644858496800x x  - 15498131256568800x x  + 38987855169056640x x  - 5829415165324800x ,
                                       0                    0 1                    0 1                   0 1                   1                    0 2                     0 1 2                    0 1 2                   1 2                    0 2                    0 1 2                    1 2                   0 2                    1 2                   2                  0 3                     0 1 3                    0 1 3                    1 3                     0 2 3                     0 1 2 3                    1 2 3                    0 2 3                     1 2 3                  2 3                    0 3                    0 1 3                    1 3                    0 2 3                    1 2 3                    2 3                    0 3                   1 3                   2 3                 3                    0 4                     0 1 4                     0 1 4                    1 4                     0 2 4                     0 1 2 4                     1 2 4                    0 2 4                     1 2 4                    2 4                     0 3 4                      0 1 3 4                     1 3 4                     0 2 3 4                     1 2 3 4                    2 3 4                   0 3 4                    1 3 4                    2 3 4                    3 4                     0 4                     0 1 4                     1 4                     0 2 4                     1 2 4                     2 4                     0 3 4                      1 3 4                     2 3 4                    3 4                     0 4                     1 4                     2 4                     3 4                    4
                      
                                         4                     3                       2 2                      3                   4                    3                      2                          2                      3                     2 2                        2                    2 2                     3                      3                   4                    3                       2                           2                       3                      2                                                    2                          2                         2                    3                       2 2                        2                    2 2                        2                        2                    2 2                       3                      3                      3                   4                    3                       2                          2                     3                     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                      3                      3                       3                    4
                      - 3757910502528000x  + 14927467023463680x x  - 17225008498014240x x  + 7181228356342320x x  - 964075013516160x  - 3167214514531200x x  + 8696473473094680x x x  - 6871698206337895x x x  + 1878970874447520x x  + 929243555163000x x  - 4882480330372659x x x  + 2803080412746096x x  + 927824064010200x x  - 2382201767370600x x  + 133134308354880x  - 1502952506899200x x  - 27470238517737120x x x  + 36193143999673920x x x  - 10509094709470560x x  - 8498705218069920x x x  + 14458571980647760x x x x  - 9197293063489440x x x  - 4132433802656124x x x  + 10271455210097844x x x  - 46981140889536x x  + 22707727246834560x x  + 3273514987444800x x x  - 7302475990261760x x  + 5574488792925360x x x  - 3069827698878920x x x  - 3139464910826160x x  - 10248193081365120x x  + 3819107241895680x x  + 4690376671676640x x  + 916151680684800x  - 7066284521909760x x  + 17231600378795040x x x  - 5421217739586740x x x  + 923119932092400x x  - 389111588117280x x x  - 10272193687828800x x x x  + 6171328188747750x x x  + 4301189326934892x x x  - 14612703900273402x x x  + 1589688234954288x x  + 14030021206521600x x x  - 20259470057866480x x x x  - 3694160127203840x x x  - 7864485146043120x x x x  + 50106017265840760x x x x  - 3867369814827288x x x  - 14363952321472320x x x  + 3320501244409440x x x  - 13424476306347120x x x  + 19194965381894400x x  - 1630793147639040x x  - 10358508530739600x x x  + 5259427945703400x x  + 8148233116371600x x x  - 19539996963893280x x x  + 6585284405274120x x  + 43163714099520x x x  + 30528380367189600x x x  - 20497433035421160x x x  - 3092224273689600x x  + 5806555444166400x x  - 8693174678328000x x  + 7545154608880560x x  - 12253359749954400x x  + 2785634858059200x ,
                                         0                     0 1                     0 1                    0 1                   1                    0 2                    0 1 2                    0 1 2                    1 2                   0 2                    0 1 2                    1 2                   0 2                    1 2                   2                    0 3                     0 1 3                     0 1 3                     1 3                    0 2 3                     0 1 2 3                    1 2 3                    0 2 3                     1 2 3                  2 3                     0 3                    0 1 3                    1 3                    0 2 3                    1 2 3                    2 3                     0 3                    1 3                    2 3                   3                    0 4                     0 1 4                    0 1 4                   1 4                   0 2 4                     0 1 2 4                    1 2 4                    0 2 4                     1 2 4                    2 4                     0 3 4                     0 1 3 4                    1 3 4                    0 2 3 4                     1 2 3 4                    2 3 4                     0 3 4                    1 3 4                     2 3 4                     3 4                    0 4                     0 1 4                    1 4                    0 2 4                     1 2 4                    2 4                  0 3 4                     1 3 4                     2 3 4                    3 4                    0 4                    1 4                    2 4                     3 4                    4
                      
                                       4                     3                       2 2                      3                   4                    3                      2                          2                      3                      2 2                        2                    2 2                      3                      3                   4                    3                     2                          2                     3                      2                                                   2                          2                       2                      3                      2 2                       2                   2 2                        2                       2                   2 2                     3                     3                     3                   4                    3                       2                           2                      3                      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                       3                       3                      3                     4
                      1796818754304000x  - 13918313285452800x x  + 11057837133067200x x  - 1367876546080800x x  - 421384286836800x  + 2345187541939200x x  + 1924863069150000x x x  - 7090386260826960x x x  + 2222362931748120x x  + 3333536181385200x x  - 5769697986217200x x x  + 4856408821182240x x  + 1437918360303600x x  - 5582937701156760x x  + 302618414901600x  + 6703841808115200x x  + 871877501772480x x x  - 1749060820368960x x x  - 122529195673920x x  - 3316514727939840x x x  - 5017928687809800x x x x  + 3261726576707760x x x  - 1347350763984120x x x  + 246792514958640x x x  + 1160896560485760x x  - 3329723351808000x x  + 662711497348320x x x  + 120605709329280x x  + 5086457930403360x x x  - 644883182546400x x x  - 184855548638880x x  + 548400127887360x x  - 333825774750720x x  - 918436488177600x x  - 115625273783040x  + 9907440734822400x x  - 57695214385132800x x x  + 41410437075050400x x x  - 8483814192722400x x  - 1445566311020160x x x  - 8220338860861680x x x x  + 12838245577177860x x x  + 538375149426240x x x  - 26585175391971840x x x  + 3223684041037740x x  + 14247508936899840x x x  - 294104884519040x x x x  + 476757297676800x x x  - 3646797810410720x x x x  - 10743926315504760x x x x  + 1976833318624920x x x  + 9129589746453120x x x  + 2547634933710720x x x  + 9012313226190960x x x  - 2065855999777920x x  + 20868739996608000x x  - 14599008595236800x x x  - 19807351115050800x x  + 2010276325422400x x x  + 28749423561106500x x x  + 8337129436171620x x  - 15209554039411200x x x  - 15644972082801600x x x  - 14899124513700000x x x  + 40374185931878400x x  - 6664517879040000x x  + 55672904162106000x x  - 19449603985165200x x  + 6205260168576000x x  - 20950401935232000x
                                       0                     0 1                     0 1                    0 1                   1                    0 2                    0 1 2                    0 1 2                    1 2                    0 2                    0 1 2                    1 2                    0 2                    1 2                   2                    0 3                   0 1 3                    0 1 3                   1 3                    0 2 3                    0 1 2 3                    1 2 3                    0 2 3                   1 2 3                    2 3                    0 3                   0 1 3                   1 3                    0 2 3                   1 2 3                   2 3                   0 3                   1 3                   2 3                   3                    0 4                     0 1 4                     0 1 4                    1 4                    0 2 4                    0 1 2 4                     1 2 4                   0 2 4                     1 2 4                    2 4                     0 3 4                   0 1 3 4                   1 3 4                    0 2 3 4                     1 2 3 4                    2 3 4                    0 3 4                    1 3 4                    2 3 4                    3 4                     0 4                     0 1 4                     1 4                    0 2 4                     1 2 4                    2 4                     0 3 4                     1 3 4                     2 3 4                     3 4                    0 4                     1 4                     2 4                    3 4                     4
                     }

o3 : RationalMap (Cremona transformation of PP^4)

See also

Ways to use this method:

The source of this document is in Cremona/documentation.m2:412:0.