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Cremona :: inverse(RationalMap)

inverse(RationalMap) -- inverse of a birational map

Synopsis

Description

inverse(phi,MathMode=>true) is the same as inverseMap(phi,MathMode=>true,Verbose=>false), while inverse(phi,MathMode=>false) applies a middle ground approach between inverseMap(phi,MathMode=>true) and inverseMap(phi,MathMode=>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: {
                        87713 4   171940669 3     786693217 2 2   1176261   3   174575 4   456216499 3     52359194927 2       5579886061   2     13389751 3     191644459 2 2   59850847787     2   831631429 2 2   31307303   3   8534795951   3   10081913 4   11675519 3     192249615263 2       36291945199   2     56838053 3     1873054933 2       292082344411           23709080881 2       74483500913   2     35210041763   2     2080929049 3     203544727 2 2   5850860317     2   53160721 2 2   34691361079     2   10467532387     2   58824896633 2 2   398325457   3   514981   3    829687   3   109219 4   87928937 3     37380364421 2       12850450633   2     304700929 3     600713887 2       520240788881           12352174879 2       192049751   2     3302801963   2     36931661 3     588577417 2       1475796509           8966991391 2       47105896363           10761984361           1540753237 2       7830532057   2     10360739819   2     126855097379   2     97816669 3     194637089 2 2   129721589     2   810200723 2 2   3458531599     2   10016990171     2   402378101 2 2   210959377967     2   409370286581     2   374657652319     2   11558520289 2 2   862261831   3   190847047   3   211570741   3   4335215   3   489173 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 ,
                         7840 0    3628800  0 1    16934400 0 1     8960  0 1    2268  1    21168000 0 2    444528000  0 1 2    39513600  0 1 2    282240  1 2   105840000 0 2    889056000  0 1 2    35562240 1 2    3307500 0 2    296352000 1 2    1764000 2    441000  0 3    1143072000  0 1 3    152409600  0 1 3    1306368 1 3   2222640000 0 2 3    1600300800  0 1 2 3    114307200  1 2 3    2222640000 0 2 3    5334336000 1 2 3    222264000 2 3    3175200  0 3    342921600 0 1 3    1224720 1 3    222264000  0 2 3    228614400  1 2 3    1333584000 2 3    4762800  0 3    30240 1 3   1058400 2 3    3780  3   19051200 0 4    304819200  0 1 4    152409600  0 1 4    7620480  1 4   127008000 0 2 4    3556224000  0 1 2 4    266716800  1 2 4    42336000 0 2 4    84672000  1 2 4    7056000 2 4    35721000 0 3 4    78382080  0 1 3 4    54867456  1 3 4    533433600  0 2 3 4     65318400  1 2 3 4    15876000  2 3 4    428652000 0 3 4     34292160  1 3 4    1333584000  2 3 4    6350400 3 4    11907000 0 4    12096000 0 1 4    15240960 1 4    63504000  0 2 4    1778112000 1 2 4    21168000 2 4    2857680000  0 3 4    2286144000  1 3 4    3333960000  2 3 4    190512000  3 4    19845000 0 4    63504000 1 4    3307500  2 4    254016 3 4   126000 4
                      
                      77687 4   1893375163 3     745990081 2 2   246515089   3   35446399 4   12828001 3     4299402481 2       117657989077   2     520789147 3     47018753 2 2   2447714863     2   14479054193 2 2   536047   3   1192105381   3   518639 4   5114703379 3     3691001057 2       79302551   2     613515787 3     6101491049 2       2497012027           24426294703 2       8032480129   2     393103933   2     980054 3     1676997047 2 2   16148725991     2   541267781 2 2   35711433817     2   1583696827     2   1235700847 2 2   87106963   3   625319561   3   298242641   3   544463 4   13856723 3     10624110179 2       48549025723   2     734224931 3     409063541 2       1819317229           98792752963 2       274596281   2     94078309523   2     122592691 3      61593467 2       1430441815217           7429733591 2       83059865963           4387536922399           22245892841 2       161480778617   2     14301904817   2     22041159893   2     261691907 3     36414451 2 2   24541197491     2   1293610919 2 2   75958487     2   49534308121     2   789360553 2 2   138709401809     2   38760394277     2   99078815873     2   7078651 2 2   3013669   3   116122337   3   320231   3   17810837   3   1247 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 ,
                      44100 0    74088000  0 1    3704400  0 1    1693440  0 1     80640  1    2940000 0 2    37044000  0 1 2     177811200  0 1 2    8467200  1 2    1470000 0 2     9878400  0 1 2     50803200  1 2    17640 0 2    14817600  1 2   117600 2    277830000 0 3    17781120  0 1 3    864000  0 1 3     907200  1 3    123480000 0 2 3     6945750  0 1 2 3    190512000  1 2 3    222264000 0 2 3    5556600  1 2 3   231525 2 3    20837250  0 3    190512000  0 1 3    10886400 1 3    666792000  0 2 3    142884000 1 2 3    66679200  2 3    4762800 0 3    8164800  1 3    9525600  2 3   113400 3    1984500 0 4    296352000  0 1 4    254016000  0 1 4    2419200  1 4    13608000 0 2 4     7056000  0 1 2 4    190512000  1 2 4    95256000 0 2 4    533433600  1 2 4    15876000 2 4   200037600 0 3 4    11430720000  0 1 3 4    11664000  1 3 4    1714608000 0 2 3 4    34292160000  1 2 3 4    444528000  2 3 4    1428840000  0 3 4     36288000  1 3 4    952560000  2 3 4    13608000 3 4   31752000 0 4    317520000  0 1 4     6048000  1 4    3402000 0 2 4    119070000  1 2 4    31752000 2 4    1428840000  0 3 4     81648000  1 3 4    476280000  2 3 4    38880  3 4    302400 0 4     756000  1 4    12096 2 4    168000  3 4    300 4
                      
                      20470217 4   6197290213 3     142427891567 2 2   1607881319   3   402659 4   6075301 3     261895079099 2       1197196974853   2     129429707 3     4720865 2 2   16381014529     2   185942825261 2 2   522507949   3   50490451   3   3948499 4   187962641 3     9589727641 2       5095947877   2     8620609 3     242438299 2       30818449           2599354883 2       299855093   2     6354704233   2     30475577 3     35776819 2 2   155423077     2   331874303 2 2   9042325657     2   29069826377     2   75778393 2 2   4296577   3   33468533   3   15349   3   31001 4   7356550399 3     271122359 2       47203852511   2     404391767 3     87849878513 2       92563384943           476223629723 2       243843073   2     34868254657   2     16067797 3     253641949 2       102150887           1308957383 2       19959639787           225792765497           8331927001 2       2381334211   2     1895122339   2     42535616623   2     157432169 3     1457325563 2 2   48181219883     2   20798623 2 2   10545160561     2   119416726721     2   6035444641 2 2   916767947     2   2861857     2   1443756259     2   5256437 2 2   670106873   3   11389349   3   17933177   3   17828999   3   19613 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 ,
                       882000  0    42336000  0 1     169344000  0 1     4838400  0 1     720  1    705600 0 2     423360000  0 1 2     1016064000  0 1 2     403200  1 2    56448  0 2     26460000  0 1 2     508032000  1 2    7056000  0 2    288000  1 2    235200 2    2646000  0 3    63504000  0 1 3    14515200  0 1 3    20736  1 3    13230000 0 2 3    4704000 0 1 2 3     5443200  1 2 3    6615000  0 2 3    127008000 1 2 3    1512000 2 3    705600  0 3     345600  0 1 3     622080  1 3    127008000 0 2 3    130636800  1 2 3    1587600 2 3    151200 0 3    373248  1 3   19200 2 3   25920 3    158760000 0 4    15876000 0 1 4     63504000  0 1 4    2419200  1 4    1905120000 0 2 4    254016000  0 1 2 4     508032000  1 2 4    76204800 0 2 4    127008000  1 2 4    1814400 2 4    3969000  0 3 4    31752000 0 1 3 4     806400   1 3 4    381024000  0 2 3 4    1143072000  1 2 3 4    285768000 2 3 4    47628000  0 3 4     6531840  1 3 4    571536000  2 3 4    1632960  3 4    31752000  0 4     63504000  0 1 4    134400  1 4     63504000  0 2 4     127008000  1 2 4    63504000  2 4    3175200  0 3 4     6720  1 3 4     4762800  2 3 4    108864 3 4    5292000  0 4    302400  1 4     98000  2 4    226800  3 4    1800 4
                      
                        1500391 4   8275752563 3     256341089 2 2   155007079   3   746581 4   97867211 3     164921515901 2       24403628009   2     888520553 3     382779157 2 2   5957276623     2   509959133 2 2   885643   3   25594339   3   60397 4   124434781 3     90257972921 2       1098062669   2     2057266549 3     2039769983 2       225727376089           2460325939 2       8187871753   2     10446122737   2      675949 3     151489834 2 2   887711869     2   165710771 2 2   432824251999     2   13643011097     2   2379244109 2 2   94290809   3   6889639   3   1258555709   3   336733 4   108579713 3     34623880909 2       1486648359659   2     3469044511 3     812299043 2       101676789773           7196517913 2       4944129079   2     111040487203   2     1960711 3     738628408877 2       29850049441           23161548511 2       350948383           500867067709           3587852269 2       35587222067   2     1638347143   2     4960572769   2     89034755 3     2840497253 2 2   185863846411     2   389117173 2 2   36324424369     2   17653930411     2   103864601 2 2   594188313743     2   2800420663     2   12401450617     2   898438523 2 2   1147560823   3   912182179   3   20621633   3   26614447   3    563 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 ,
                         211680 0    88905600  0 1    3704400  0 1    5080320  0 1    6480  1    3528000 0 2     889056000  0 1 2    106686720  0 1 2    38102400 1 2    10584000 0 2    74088000  0 1 2    5292000  1 2    49000 0 2    1411200 1 2   12000 2    2778300  0 3    1333584000 0 1 3    31752000  0 1 3     5443200  1 3    22226400  0 2 3     833490000  0 1 2 3    63504000  1 2 3    222264000 0 2 3     90720000  1 2 3   3528000 2 3    10418625 0 3    63504000 0 1 3     604800  1 3    6667920000  0 2 3    114307200  1 2 3    119070000 2 3    4762800 0 3    466560 1 3    95256000  2 3   226800 3    95256000 0 4    889056000  0 1 4     5334336000  0 1 4    38102400  1 4    35721000 0 2 4    5715360000  0 1 2 4    381024000 1 2 4    95256000  0 2 4     952560000  1 2 4    84000  2 4    10001880000 0 3 4    228614400  0 1 3 4    228614400  1 3 4    3499200  0 2 3 4    1714608000  1 2 3 4    105840000 2 3 4    8001504000 0 3 4    76204800  1 3 4    381024000 2 3 4    3048192 3 4    272160000 0 4     889056000  0 1 4    3048192  1 4    1905120000 0 2 4     95256000  1 2 4    3024000  2 4    8001504000  0 3 4    28576800  1 3 4    190512000  2 3 4    38102400 3 4    63504000  0 4    12700800 1 4    604800  2 4    846720  3 4   2800 4
                      
                        25197 4   140616709 3     13520392891 2 2   13683556193   3   43193719 4   1310991 3     13938392243 2       20490507127   2     1092604379 3     25878779 2 2   61718231     2   17113948007 2 2   3781009   3   1399587607   3   3213307 4   201997 3     4300057681 2       18208353301   2     2892598211 3     958803161 2       8509111559           95425957337 2       8879315339   2     123111557327   2     804820927 3     838659919 2 2   64213060183     2   30588325141 2 2   9844329091     2   3831602699     2   421910291 2 2   1356293   3   90385021   3   1080170969   3   240439 4   708417343 3     69211419139 2       821880229741   2     8234269157 3     17527342831 2       5079418246169           271977807167 2       3766010893   2     73836492767   2     6083219 3     69268081709 2       166294514647           188202892601 2       30929665997           15654734391433           584160397 2       590797374319   2     395965971767   2     491764815781   2     1214588279 3     2308919491 2 2   437921877679     2   40210722691 2 2   2916578449     2   395005523113     2   315679271 2 2   534410724893     2   69884537     2   216151621817     2   2988196411 2 2   97251001   3   848159597   3   194830663   3   276462167   3   52207 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
                         2800 0    6350400  0 1     50803200  0 1     33868800  0 1    181440  1    196000 0 2    381024000  0 1 2     47628000  0 1 2    10160640  1 2    2268000 0 2    2126250 0 1 2    762048000  1 2    992250 0 2    31752000  1 2    529200 2    73500 0 3    28576800  0 1 3     32659200  0 1 3     6531840  1 3    15876000 0 2 3    21432600  0 1 2 3    137168640  1 2 3    158760000 0 2 3     571536000  1 2 3    95256000 2 3    15876000 0 3    228614400  0 1 3     65318400  1 3    190512000 0 2 3    95256000  1 2 3    31752000 2 3    352800 0 3    604800  1 3    31752000  2 3    15120 3    47628000 0 4    444528000  0 1 4    4267468800  0 1 4    152409600 1 4    714420000  0 2 4    16003008000  0 1 2 4    1524096000  1 2 4    95256000  0 2 4    533433600  1 2 4    211680 2 4    1000188000 0 3 4    2667168000  0 1 3 4     457228800  1 3 4    240045120  0 2 3 4     48009024000  1 2 3 4    14817600 2 3 4    4000752000  0 3 4    1524096000  1 3 4    2667168000  2 3 4    27216000  3 4    190512000 0 4    1778112000  0 1 4    304819200  1 4    47628000  0 2 4    1333584000  1 2 4    5292000  2 4    4000752000  0 3 4   71442000 1 3 4    1333584000  2 3 4    95256000  3 4    3969000 0 4   127008000 1 4    5292000  2 4    21168000 3 4   28000 4
                     }

o2 : RationalMap (rational map from PP^4 to PP^4)
i3 : time inverse phi
     -- used 0.0975358 seconds

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
                      - 122896266822000x  + 610226014948200x x  - 576753063780600x x  + 102720879015300x x  - 9414842673600x  + 3699852055200x x  + 294739615170000x x x  - 138153302961660x x x  - 20531975385600x x  + 67442944887360x x  + 72946285062840x x x  - 22536566332560x x  - 20227694518080x x  + 53474573648160x x  - 19027177344000x  + 543647816472600x x  - 729632501047800x x x  + 393880976764800x x x  + 17332571210850x x  + 138326986841400x x x  - 239573496489940x x x x  - 50612943405150x x x  + 76933021250736x x x  - 76938251041620x x x  + 61524106237440x x  + 388020756853800x x  - 454982331858600x x x  + 275439328969350x x  + 335496966653240x x x  - 334524418004700x x x  + 74995973730720x x  - 571931229844800x x  + 411337054789500x x  - 177295703054400x x  - 72053361828000x  - 470991675558600x x  + 684524587950000x x x  - 343387979006100x x x  + 111129455341800x x  - 52885876241520x x x  + 227503935136240x x x x  - 265242788438040x x x  + 21164925227040x x x  + 182800484601840x x x  - 42083923896000x x  + 998536378213080x x x  - 1064263857132840x x x x  + 603623670792450x x x  + 676716868932748x x x x  - 626804085414660x x x x  + 163725370837560x x x  - 160427754360720x x x  + 340388071535400x x x  - 22401861485640x x x  - 3623047936200x x  - 426737044818000x x  + 227827854626400x x x  - 49334139653400x x  - 92973815620560x x x  + 10918769215200x x x  + 26397954424800x x  + 757906867892160x x x  - 153494997056100x x x  + 17739208087440x x x  + 50625488377800x x  - 271153558821600x x  + 8845812435600x x  + 7986790425600x x  + 42802362262800x x  - 42149700432000x ,
                                        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
                      15912286185600x  - 32666729760000x x  - 15108013053000x x  + 15508549507500x x  - 3635299974600x  - 30053222277120x x  + 67143621337920x x x  - 45819417514320x x x  + 3125261592180x x  - 30415524074496x x  + 27467429978016x x x  - 11013043021920x x  + 7978479813504x x  - 7818421145760x x  + 6786624268800x  + 2852620848000x x  - 54380379049920x x x  - 14059142854260x x x  + 9343146891960x x  + 79896990438720x x x  + 7987739976640x x x x  - 25646613872940x x x  + 19021075914240x x x  + 32507048522880x x x  - 9037822798080x x  + 41170201541280x x  - 129277287708240x x x  + 100692437922720x x  + 67019128295392x x x  - 134970011087280x x x  + 26495069231232x x  - 1425580880640x x  + 89454000031200x x  - 9264577136640x x  - 79671916598400x  + 12396780012960x x  + 64270393367040x x x  - 60099940693380x x x  + 12129811669080x x  + 2300911411392x x x  + 23325405967136x x x x  - 6082343882748x x x  + 73111088663712x x x  - 68120486026992x x x  + 41148725837760x x  - 98159593245840x x x  + 156513277197480x x x x  + 16501520276640x x x  - 207759315898960x x x x  + 195854027927160x x x x  - 96637565423232x x x  + 312482519545056x x x  - 133557694053840x x x  + 116796604427616x x x  + 37928741981760x x  - 91936123478640x x  + 29143098434520x x x  + 16869404149920x x  + 27342751884480x x x  - 83406790985280x x x  + 51118373655360x x  - 192278537972640x x x  + 142531521945120x x x  - 88890206851680x x x  + 9031252991040x x  - 63958637836800x x  + 15433367065200x x  - 21116429006400x x  + 38945764872000x x  - 22590246168000x ,
                                     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
                      - 52294658119200x  - 73775781752400x x  + 377375404858200x x  - 74465712770400x x  + 4404619800000x  - 54482531040000x x  - 40382993229840x x x  + 65390796852150x x x  + 18463537352400x x  + 2252721466800x x  - 47125565704560x x x  + 26983125040200x x  + 21432159826560x x  - 37470504057600x x  + 6994536192000x  + 260594623850400x x  + 290841609214800x x x  - 303850711350900x x x  - 9778848013200x x  - 29582678880720x x x  - 5821672506720x x x x  + 26932109457575x x x  - 114157548135120x x x  + 62244094514000x x x  - 12743685696000x x  + 56768113573920x x  + 116528888316960x x x  - 132283849065900x x  - 177817600243080x x x  + 204495419762900x x x  - 122915918681920x x  - 244398593890080x x  - 106437100652400x x  + 63842392588800x x  + 263445795043200x  + 68901019210800x x  - 372047414193000x x x  + 150903886102200x x x  - 87019431651600x x  + 258978077139600x x x  - 115436174046840x x x x  + 199039436315860x x x  + 128225836841760x x x  - 133218587957360x x x  + 10342556582400x x  - 794753437625280x x x  + 257762615322240x x x x  - 505293200605200x x x  - 806213859353028x x x x  + 400237544346010x x x x  - 11414117909600x x x  + 126206635227120x x x  - 267310968237600x x x  - 287360473692600x x x  + 32167828425600x x  + 541064472429600x x  - 276072673766400x x x  + 68463563968800x x  + 473384816195760x x x  - 89147044933800x x x  - 5338173120000x x  - 1436138162166960x x x  + 246331058941800x x x  - 66870834534840x x x  - 28553692305600x x  + 553638058041600x x  - 55469319393600x x  + 48823967858400x x  - 154582777792800x x  + 90765505152000x ,
                                       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
                      225299805544800x  - 330642588691800x x  + 27235893138300x x  + 4343860521000x x  + 1629779356800x  + 159823007245440x x  - 30763120793040x x x  + 53087799420x x x  + 2357693352000x x  - 57993313351968x x  + 87594730201440x x x  - 11601915422400x x  - 46757080311840x x  + 11671845984000x x  - 3195942048000x  + 39541091058000x x  - 120804577464060x x x  + 28835014157460x x x  - 2466278992800x x  - 11861418547440x x x  + 52491339201900x x x x  - 35328936789360x x x  - 72943723015680x x x  + 69884569051200x x x  - 28497367821600x x  - 201958009011360x x  + 305966037223980x x x  - 47053010390400x x  - 71004284369424x x x  - 16114742635500x x x  + 23260801016880x x  + 114072546052080x x  - 109248978787200x x  + 70993732340160x x  - 15491192749200x  - 275388970968720x x  + 62625658942980x x x  + 53255457148320x x x  + 29163843465120x x  - 576912053785824x x x  + 404199969170424x x x x  - 67449368795760x x x  - 266641184600160x x x  + 58294033645680x x x  - 13960444449600x x  + 160714167148080x x x  - 12916558170180x x x x  - 26261077229100x x x  - 94356272818080x x x x  + 119998878313770x x x x  - 90778928140440x x x  + 126390462812208x x x  - 59979448449900x x x  + 58653041453280x x x  - 2420069830920x x  - 845779476744720x x  + 628310108510280x x x  - 78558663501360x x  - 582730586356320x x x  + 181873147979880x x x  - 65589467119200x x  + 94124711908080x x x  - 61731213131700x x x  + 13912561897560x x x  + 15055109125560x x  - 232983267451200x x  + 121634372487600x x  - 77210531143200x x  + 11083813297200x x  - 33720624000x ,
                                      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
                      26800218446400x  - 69315776006400x x  - 7878027748500x x  - 4717267948050x x  + 13887758556000x  - 9134842500000x x  - 187370537916240x x x  + 237359616229120x x x  - 22502882833200x x  - 15239563197696x x  - 133834103989600x x x  + 5472770520000x x  + 13610153329920x x  - 10555926360000x x  + 7196874624000x  - 195612212628000x x  + 261387739630080x x x  - 157547757764460x x x  - 55942226460225x x  - 255570602711760x x x  + 291230119915840x x x x  + 161137875687600x x x  - 43253689984960x x x  - 117978688008720x x x  + 12055394668800x x  - 5907824748000x x  + 156082348994880x x x  - 256381925276925x x  - 254719525231920x x x  + 366271610224800x x x  - 71709385681920x x  + 387462921665400x x  - 219335874968700x x  + 15303220430400x x  + 59254693344000x  + 185730473214720x x  - 495104912099280x x x  + 307458192136410x x x  - 40089460842900x x  + 127821373553232x x x  - 305764104788352x x x x  + 54154262364720x x x  - 526481441280x x x  - 1939032443040x x x  - 298224547200x x  - 494696985166440x x x  + 14347439336160x x x x  + 92012951496375x x x  + 134677601441280x x x x  - 195719283432120x x x x  + 86753997181920x x x  - 860450603570460x x x  + 315809652088350x x x  - 155220929307360x x x  + 4037540207400x x  + 364641534689040x x  - 311216066852040x x x  + 92748263687100x x  + 130329065616480x x x  - 86447865818160x x x  + 8594176291200x x  + 62272611138780x x x  + 53164776290850x x x  - 25036510773600x x x  - 9045236919600x x  + 76794624471600x x  - 38904748432200x x  + 20464559539200x x  + 7610584602600x x  - 3236751252000x
                                     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