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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 0.0174462 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 1.06625 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
                      - 504829t  + 3229176t t  - 4562550t  - 4459889t t  + 4322859t t  + 4902406t  - 2392187t t  - 3546510t t  - 3444057t t  - 753699t  - 476139t t  + 10192t t  - 2986367t t  - 3950341t t  - 1133935t  + 1674708t t  - 618804t t  + 315841t t  + 3155923t t  - 4313726t t  + 60196t  + 4436333t t  + 445658t t  - 938211t t  + 3387775t t  - 1601930t t  + 2461433t t  + 2001752t ,
                               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
                      1487252t  - 4924428t t  - 1911703t  - 4130820t t  + 1545559t t  + 4421923t  + 3567211t t  + 151525t t  + 2462682t t  - 3670699t  - 4346000t t  - 1983008t t  + 2957070t t  + 2851548t t  + 1377348t  + 702578t t  + 3376029t t  - 2471380t t  + 2274186t t  + 517265t t  + 2995696t  + 4601592t t  + 698380t t  + 2407012t t  - 4145247t t  + 1698509t t  - 2464618t t  - 899103t ,
                              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
                      - 2645998t  + 4124448t t  + 4335664t  + 1138774t t  + 4218178t t  + 1301734t  - 3894435t t  + 2555998t t  - 858551t t  - 4322407t  - 3980348t t  - 4739225t t  + 3899459t t  - 4585874t t  + 3285060t  - 26077t t  + 4961529t t  + 2382243t t  - 1905104t t  - 1983390t t  + 491233t  + 430607t t  + 2434946t t  - 2286649t t  - 2697467t t  - 3922195t t  + 4361600t t  - 2508443t ,
                                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
                      1300350t  + 2375t t  - 1463923t  + 3248345t t  + 1693747t t  + 281578t  - 2558052t t  + 2827624t t  + 2037235t t  - 4016977t  - 1398979t t  + 631179t t  + 4656744t t  + 814120t t  - 3670003t  - 2303918t t  + 3722991t t  - 841659t t  - 4664497t t  + 4502747t t  + 992386t  - 1456791t t  - 2042982t t  - 378180t t  + 4447876t t  - 90777t t  + 4080602t t  + 3563837t ,
                              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
                      1755735t  - 1232809t t  + 3138541t  - 1582940t t  - 1359259t t  - 3588226t  + 4754923t t  + 2082709t t  - 4025615t t  - 2755942t  - 639738t t  - 2076498t t  + 3568331t t  - 4065981t t  + 1925954t  + 4128618t t  - 1249725t t  + 1087035t t  - 3182438t t  - 2987997t t  - 3983194t  - 4239316t t  + 2139714t t  - 4980249t t  - 4787383t t  - 3288221t t  + 4158613t t  - 1979785t ,
                              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
                      3230006t  + 3773089t t  + 4332553t  - 4142082t t  + 275239t t  - 4379770t  + 4980733t t  - 3884484t t  - 1147849t t  - 623821t  - 3264698t t  + 4083082t t  + 4531652t t  - 3688165t t  - 3441687t  - 4504438t t  + 2858329t t  - 4521790t t  - 1431400t t  - 2378524t t  + 700787t  - 3989391t t  + 4499010t t  + 4247848t t  + 3510772t t  - 2774435t t  - 254652t t  - 296603t ,
                              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
                      - 1857074t  - 513714t t  + 3323746t  + 4020863t t  - 1813353t t  + 1020751t  + 790421t t  - 2328028t t  + 656053t t  - 1629478t  - 3393663t t  + 1757333t t  + 2737318t t  - 339621t t  + 128767t  - 4615880t t  + 710105t t  + 2112127t t  - 4031886t t  + 4790173t t  + 1795943t  - 3880433t t  + 3253627t t  + 213558t t  + 1444271t t  - 3191486t t  - 2879596t t  + 3402237t ,
                                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
                      - 2899302t  - 2622888t t  - 1114117t  + 725869t t  + 775021t t  - 3333412t  + 3111511t t  + 3014162t t  - 2082844t t  - 1098013t  + 3824880t t  - 3242917t t  + 4760521t t  + 320501t t  - 3998075t  - 3302670t t  - 645279t t  - 2798786t t  + 4363171t t  + 1874533t t  + 3379649t  + 2260922t t  - 1691718t t  - 2021154t t  - 718336t t  + 753075t t  + 1406452t t  - 2063190t ,
                                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
                      - 1176583t  + 11786t t  + 262829t  - 3797888t t  + 3876083t t  + 1979905t  - 3825529t t  - 975389t t  - 4516356t t  - 3076034t  - 2980490t t  - 1101683t t  + 4367802t t  + 3785614t t  + 2223475t  - 1377679t t  + 1138586t t  - 4962588t t  - 3324451t t  + 2323683t t  + 4384016t  + 2632722t t  - 3213844t t  + 692945t t  + 1150283t t  + 29776t t  + 1372978t t  - 210547t ,
                                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 :