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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 0.0218045 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 0.391822 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
                      3595775t  + 1955085t t  - 3216200t  + 4854122t t  + 1205502t t  - 1975634t  + 3832790t t  - 1340758t t  - 2876523t t  - 2376974t  + 413617t t  + 697370t t  + 1458127t t  - 178971t t  + 1598242t  + 2885494t t  - 3431573t t  - 2594430t t  - 4118744t t  + 920588t t  - 928298t  - 1355233t t  - 2841870t t  + 4503191t t  + 4578965t t  - 4703003t t  - 2255578t t  + 3369687t ,
                              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
                      196923t  + 735740t t  + 4204361t  + 4919087t t  + 4191608t t  - 4897776t  - 80313t t  + 1616507t t  + 2913552t t  - 661472t  - 1950679t t  + 1466426t t  + 4735302t t  - 4114190t t  - 573844t  + 3471476t t  + 713534t t  - 3588772t t  + 4356229t t  - 1094071t t  + 1819413t  + 2855848t t  + 673382t t  - 2207004t t  - 4313805t t  + 2594471t t  + 1797216t t  + 4464199t ,
                             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
                      - 4465202t  + 2675358t t  + 2673408t  + 1608117t t  + 455149t t  + 1414960t  - 1918352t t  - 2585719t t  + 3705854t t  + 4858205t  + 4123629t t  + 2628647t t  + 3545835t t  - 323076t t  - 3931135t  + 2617377t t  + 2924145t t  - 359942t t  + 518309t t  - 4870860t t  + 1668542t  - 1288445t t  + 4649334t t  - 4911574t t  - 4445126t t  + 312193t t  + 3435991t t  + 3339675t ,
                                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
                      730764t  - 3118985t t  - 633600t  + 1675759t t  - 1983772t t  - 3699329t  - 2028122t t  + 624554t t  + 2269469t t  - 122152t  - 4405808t t  + 1143542t t  - 2920885t t  - 3170637t t  + 4089506t  + 704014t t  - 2749909t t  - 2413980t t  + 2026520t t  - 4599448t t  - 3928920t  - 2294470t t  - 4750280t t  + 14950t t  - 1091822t t  + 3402770t t  + 4310641t t  + 1480834t ,
                             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
                      - 4315118t  - 3098978t t  - 4885672t  - 359755t t  + 278623t t  - 1652496t  - 759743t t  - 3302491t t  - 4526561t t  + 4004742t  - 350777t t  - 1379421t t  + 70322t t  - 4746577t t  - 1946552t  + 4575853t t  + 3252519t t  - 1481191t t  + 368114t t  - 1788374t t  - 129069t  - 1206452t t  - 1240321t t  - 913163t t  - 2720448t t  - 3441947t t  + 2765564t t  - 17463t ,
                                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
                      4437767t  + 2448998t t  - 4674501t  + 2014093t t  + 3178605t t  + 4120464t  + 1117090t t  - 700111t t  + 2370933t t  + 3414676t  - 3049078t t  - 1232613t t  + 2982730t t  - 1314587t t  - 1142231t  - 3870882t t  + 4164403t t  - 3311199t t  + 982524t t  + 4166497t t  + 2308221t  - 3559246t t  - 2674453t t  + 82769t t  - 2512434t t  - 2311487t t  + 4707485t t  - 988168t ,
                              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
                      - 437502t  - 3720847t t  - 4825935t  - 527348t t  + 4523783t t  - 1306564t  - 3597282t t  - 3264013t t  - 35115t t  - 97631t  + 570254t t  + 1228457t t  - 4317755t t  - 4288159t t  + 1658388t  - 2526156t t  - 83660t t  + 3701572t t  - 861625t t  + 1939090t t  + 273554t  - 1207486t t  - 3084758t t  - 3174489t t  + 4738002t t  - 1560900t t  + 1883185t t  - 1461027t ,
                               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
                      723816t  + 4353780t t  + 4582020t  + 4681410t t  - 2761246t t  - 1717794t  + 4053489t t  + 4253348t t  + 4652620t t  + 4003848t  + 4173348t t  - 1551664t t  - 3977382t t  - 351623t t  - 2947146t  - 3378384t t  + 61740t t  + 369478t t  + 4209042t t  - 945255t t  - 2702875t  + 2487074t t  - 2645391t t  - 1291715t t  + 2838290t t  - 271417t t  - 4299261t t  + 4539436t ,
                             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
                      1105636t  + 884462t t  + 858814t  - 314377t t  - 1521147t t  - 3344493t  + 4782364t t  - 33386t t  + 4054274t t  - 2734383t  - 4083729t t  - 3802210t t  - 2325347t t  + 4622529t t  + 4519057t  + 4236565t t  - 4508343t t  + 1442297t t  + 4717591t t  - 2796257t t  + 2305249t  - 1559784t t  + 3348653t t  + 1454428t t  + 2907444t t  + 4195869t t  - 582563t t  + 3805702t ,
                              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 :