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Cremona :: parametrize

parametrize -- parametrization of linear varieties and hyperquadrics

Synopsis

Description

i1 : P9 := ZZ/10000019[x_0..x_9]

        ZZ
o1 = --------[x , x , x , x , x , x , x , x , x , x ]
     10000019  0   1   2   3   4   5   6   7   8   9

o1 : PolynomialRing
i2 : L = trim ideal(random(1,P9),random(1,P9),random(1,P9),random(1,P9))

o2 = ideal (x  - 1112016x  - 3901361x  - 3193863x  + 4143040x  - 1964417x  +
             3           4           5           6           7           8  
     ------------------------------------------------------------------------
     1074958x , x  + 632284x  + 492458x  + 3869254x  + 2840266x  + 4883974x 
             9   2          4          5           6           7           8
     ------------------------------------------------------------------------
     + 3340961x , x  + 4724709x  - 3505386x  + 2469206x  - 1381515x  +
               9   1           4           5           6           7  
     ------------------------------------------------------------------------
     2331280x  - 4936229x , x  - 2094456x  - 3936498x  - 4665404x  - 736943x 
             8           9   0           4           5           6          7
     ------------------------------------------------------------------------
     - 849671x  + 3034137x )
              8           9

                 ZZ
o2 : Ideal of --------[x , x , x , x , x , x , x , x , x , x ]
              10000019  0   1   2   3   4   5   6   7   8   9
i3 : time parametrize L
     -- used 0.00454595 seconds

o3 = -- rational map --
                     ZZ
     source: Proj(--------[t , t , t , t , t , t ])
                  10000019  0   1   2   3   4   5
                     ZZ
     target: Proj(--------[x , x , x , x , x , x , x , x , x , x ])
                  10000019  0   1   2   3   4   5   6   7   8   9
     defining forms: {
                      2094456t  + 3936498t  + 4665404t  + 736943t  + 849671t  - 3034137t ,
                              0           1           2          3          4           5
                      
                      - 4724709t  + 3505386t  - 2469206t  + 1381515t  - 2331280t  + 4936229t ,
                                0           1           2           3           4           5
                      
                      - 632284t  - 492458t  - 3869254t  - 2840266t  - 4883974t  - 3340961t ,
                               0          1           2           3           4           5
                      
                      1112016t  + 3901361t  + 3193863t  - 4143040t  + 1964417t  - 1074958t ,
                              0           1           2           3           4           5
                      
                      t ,
                       0
                      
                      t ,
                       1
                      
                      t ,
                       2
                      
                      t ,
                       3
                      
                      t ,
                       4
                      
                      t
                       5
                     }

o3 : RationalMap (linear rational map from PP^5 to PP^9)
i4 : Q = trim ideal(random(2,P9),random(1,P9),random(1,P9))

                                                                           
o4 = ideal (x  - 3731285x  + 569485x  + 4255201x  - 2098712x  - 4248990x  -
             1           2          3           4           5           6  
     ------------------------------------------------------------------------
                                                                    
     1801342x  + 4050229x  - 2319263x , x  - 3094689x  - 4410186x  +
             7           8           9   0           2           3  
     ------------------------------------------------------------------------
                                                                            2
     3196146x  + 2713771x  + 2261412x  - 1267196x  - 4210403x  + 285932x , x 
             4           5           6           7           8          9   2
     ------------------------------------------------------------------------
                            2                                       2  
     + 1045421x x  + 718532x  - 3701628x x  + 3903798x x  + 2842397x  -
               2 3          3           2 4           3 4           4  
     ------------------------------------------------------------------------
                                                       2                
     2997962x x  + 4189835x x  + 1489225x x  - 2279955x  + 2520782x x  +
             2 5           3 5           4 5           5           2 6  
     ------------------------------------------------------------------------
                                                      2                
     4494280x x  + 3101255x x  - 681950x x  + 1307490x  + 2690767x x  +
             3 6           4 6          5 6           6           2 7  
     ------------------------------------------------------------------------
                                                                    2  
     4503651x x  + 1762528x x  + 137682x x  - 2229093x x  - 4018967x  +
             3 7           4 7          5 7           6 7           7  
     ------------------------------------------------------------------------
                                                                          
     4536117x x  - 2541309x x  + 3810968x x  - 4208194x x  - 1643560x x  +
             2 8           3 8           4 8           5 8           6 8  
     ------------------------------------------------------------------------
                           2                                            
     3330573x x  - 2280516x  - 1532056x x  + 1883935x x  + 1887667x x  +
             7 8           8           2 9           3 9           4 9  
     ------------------------------------------------------------------------
                                                                     2
     1211601x x  - 2168594x x  - 1801762x x  + 3022242x x  + 3618789x )
             5 9           6 9           7 9           8 9           9

                 ZZ
o4 : Ideal of --------[x , x , x , x , x , x , x , x , x , x ]
              10000019  0   1   2   3   4   5   6   7   8   9
i5 : time parametrize Q
     -- used 0.377115 seconds

o5 = -- rational map --
                     ZZ
     source: Proj(--------[t , t , t , t , t , t , t ])
                  10000019  0   1   2   3   4   5   6
                     ZZ
     target: Proj(--------[x , x , x , x , x , x , x , x , x , x ])
                  10000019  0   1   2   3   4   5   6   7   8   9
     defining forms: {
                                2                         2                                      2                                                    2                                                                   2                                                                                2                                                                                             2
                      - 3919443t  - 4515761t t  - 4802607t  + 769772t t  + 1742271t t  - 1038543t  + 3799724t t  - 1149614t t  - 950065t t  - 1869735t  + 3129608t t  + 3268176t t  - 2460210t t  - 2164949t t  + 2844186t  - 1998621t t  + 4603959t t  - 3573200t t  + 2622867t t  - 1309755t t  - 854520t  - 1942100t t  + 4313071t t  + 677243t t  + 1205223t t  + 2969838t t  - 3992471t t  - 573413t ,
                                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
                      
                             2                         2                                       2                                                   2                                                                 2                                                                               2                                                                                              2
                      339676t  - 1843502t t  - 3515859t  - 1881741t t  - 1435813t t  + 3853220t  - 1591439t t  + 2222028t t  - 831231t t  + 506515t  + 1977134t t  - 318964t t  + 3502926t t  - 1501752t t  + 381060t  - 2631833t t  + 2915500t t  + 1965604t t  + 307902t t  - 4753876t t  - 928778t  - 710730t t  + 1441791t t  + 2220746t t  - 2073572t t  + 2499905t t  + 4037923t t  - 3144492t ,
                             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
                      
                                2                         2                                       2                                                    2                                                               2                                                                                 2                                                                                              2
                      - 3797516t  - 3261835t t  + 4469990t  + 2612518t t  + 1089371t t  - 1898627t  - 2905764t t  - 3313431t t  + 2328126t t  + 528460t  + 3284231t t  - 2410627t t  + 269261t t  + 3157218t t  - 9101t  + 3173097t t  - 3940755t t  + 4104247t t  + 1427249t t  - 2110993t t  - 3512949t  - 3198809t t  + 3708441t t  + 2235127t t  + 446685t t  - 1973588t t  + 1189419t t  + 1546877t ,
                                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
                      
                              2                       2                                       2                                                    2                                                                  2                                                                                 2                                                                                              2
                      2909824t  + 704743t t  + 683148t  + 4039608t t  + 4553303t t  + 4644113t  - 3502175t t  - 3106852t t  + 949266t t  + 3198523t  - 2016334t t  - 2762895t t  + 472212t t  - 1329029t t  - 1786955t  - 3245058t t  + 4306055t t  - 4704077t t  - 1128640t t  + 3866702t t  - 4650216t  + 2782612t t  - 2204016t t  + 3402448t t  + 368687t t  - 1382880t t  - 4282016t t  + 4292390t ,
                              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
                      
                              2                         2                                      2                                                     2                                                                   2                                                                                 2                                                                                         2
                      2130730t  + 4534219t t  + 1261110t  - 1200141t t  + 866465t t  + 4734784t  + 1670123t t  + 3982546t t  - 3704928t t  - 1869595t  - 1147417t t  + 2887743t t  - 1274150t t  + 2769050t t  + 3563916t  + 1781510t t  + 4911359t t  - 4333786t t  - 4108403t t  + 2287250t t  - 1441305t  - 3468167t t  + 4126t t  + 238325t t  - 2048349t t  + 3544421t t  + 2226516t t  - 77164t ,
                              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
                      
                                2                       2                                       2                                                   2                                                                  2                                                                                 2                                                                                              2
                      - 3359480t  - 63377t t  + 1993010t  + 1690141t t  - 3917183t t  + 3331914t  + 487752t t  + 1518121t t  + 4731396t t  - 811904t  - 4806382t t  - 2532717t t  - 909302t t  + 2243484t t  + 4638897t  - 2411777t t  + 4685691t t  - 2238070t t  - 3323904t t  + 2734300t t  + 1515599t  - 3156194t t  - 4041114t t  - 2810029t t  - 2851810t t  + 3835177t t  - 113258t t  + 4866994t ,
                                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
                      
                              2                         2                                       2                                                    2                                                                   2                                                                               2                                                                                               2
                      1742205t  - 4742261t t  - 4734909t  + 3426242t t  + 1139456t t  - 1055499t  - 1629160t t  - 3200359t t  - 1491054t t  - 777812t  + 4042731t t  - 2394359t t  - 1413747t t  - 4259569t t  + 1331666t  + 3493892t t  + 2150329t t  + 2815419t t  - 2481194t t  - 4202417t t  + 53909t  + 3315142t t  + 2245426t t  - 2139703t t  + 3090805t t  - 4348058t t  + 3457936t t  - 4712981t ,
                              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
                      
                              2                       2                                      2                                                   2                                                                  2                                                                                2                                                                                              2
                      3913925t  + 344157t t  + 284425t  - 955585t t  - 3573346t t  + 2638272t  - 3975919t t  - 199465t t  - 769771t t  - 3450281t  - 1790509t t  - 1485590t t  - 930127t t  + 2641206t t  - 3998191t  - 927224t t  + 2323166t t  - 2305340t t  - 1015315t t  + 4353120t t  - 4376126t  - 2408746t t  - 1897589t t  + 2602224t t  - 414496t t  + 1313611t t  - 2789292t t  + 1550914t ,
                              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
                      
                                2                       2                                       2                                                   2                                                                  2                                                                              2                                                                                             2
                      - 2570676t  - 360063t t  - 422687t  + 1146156t t  + 1539218t t  + 4015746t  - 3399631t t  + 974029t t  + 610022t t  + 3532378t  - 4663382t t  + 3973301t t  + 3770101t t  + 586024t t  - 4135569t  - 692081t t  - 3642736t t  + 173656t t  + 1926543t t  + 1500356t t  - 968137t  - 1636697t t  + 4613837t t  + 1064895t t  - 4517045t t  + 3208482t t  - 47442t t  + 1267079t ,
                                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
                      
                       2                        2                                       2                                                   2                                                                2                                                                                2                                                                                              2
                      t  + 2557201t t  + 897582t  - 1955667t t  - 4162832t t  + 2917974t  - 2414045t t  - 3133456t t  + 130269t t  - 921025t  + 4924767t t  + 268844t t  - 1179881t t  - 1984252t t  + 85259t  - 989568t t  - 2303912t t  - 4074596t t  - 1741629t t  - 1251685t t  - 2195520t  + 1090392t t  - 4221350t t  - 3633875t t  + 286385t t  - 4128258t t  - 2812768t t  - 2652542t
                       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
                     }

o5 : RationalMap (quadratic rational map from PP^6 to PP^9)

Ways to use parametrize :