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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 ]
     10000019  0   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 ]
              10000019  0   9
i3 : time parametrize L
     -- used 0.0322788 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 ]
              10000019  0   9
i5 : time parametrize Q
     -- used 0.362841 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
                      817998t  + 1329216t t  - 1114182t  + 1344821t t  - 1780494t t  + 2360961t  - 230018t t  + 4128916t t  - 421002t t  + 4823510t  + 1902430t t  + 3436483t t  + 1535528t t  - 1083587t t  + 1558976t  - 3106890t t  - 1827455t t  + 4953911t t  + 3668093t t  - 3881277t t  + 2443307t  + 4514889t t  - 2074478t t  + 4328893t t  - 610448t t  - 2767549t t  - 3890948t t  + 1330950t ,
                             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
                      2699292t  - 913286t t  + 1308567t  + 3742165t t  + 2234748t t  - 1090847t  + 24640t t  + 3357238t t  + 3401451t t  - 690691t  - 1092420t t  - 2920363t t  + 2706944t t  + 4626949t t  - 1500638t  - 176369t t  - 2192329t t  - 2325489t t  + 1182952t t  + 716259t t  + 2024145t  + 810232t t  - 4634403t t  - 1079647t t  - 4729356t t  - 2620971t t  + 3524740t t  - 1796321t ,
                              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
                      2409725t  - 528341t t  + 1675402t  - 707616t t  - 1281806t t  + 1537300t  - 3535002t t  - 2913175t t  + 1869596t t  - 2751246t  - 654474t t  - 3900969t t  - 2177290t t  + 4911566t t  - 2042755t  - 2218098t t  - 3795818t t  + 2809316t t  + 1031971t t  - 569887t t  + 620121t  - 130126t t  - 3301437t t  - 2796797t t  - 2217084t t  + 3651358t t  - 4490638t t  - 4626378t ,
                              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
                      - 3257377t  - 1895252t t  + 2592730t  + 2601832t t  - 2552127t t  - 3334850t  - 2546231t t  - 4666038t t  - 3274793t t  - 2571387t  - 2276758t t  - 4598320t t  - 70152t t  + 2150293t t  - 177975t  - 81305t t  - 4300106t t  + 76999t t  + 1804125t t  + 2183565t t  + 2762962t  - 1073345t t  + 2273000t t  - 1644197t t  - 880408t t  - 2890547t t  - 1618701t t  + 2885745t ,
                                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
                      366446t  + 4297213t t  + 4708643t  + 2746493t t  - 3014690t t  - 967835t  + 4546708t t  - 235720t t  + 3940703t t  + 4714119t  - 2260772t t  - 3378764t t  - 4247617t t  + 173136t t  + 2760158t  - 2546350t t  + 2267342t t  + 3767051t t  - 1767935t t  - 4090037t t  + 2006706t  - 972351t t  - 3883009t t  + 4831131t t  + 4438324t t  - 756786t t  + 4375859t t  - 1558913t ,
                             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
                      163273t  - 1180075t t  - 2150532t  + 2988598t t  + 4385649t t  - 4536315t  + 2379600t t  - 312013t t  - 629976t t  + 1770897t  - 45361t t  + 2486846t t  - 2344497t t  - 4161253t t  + 466259t  + 570919t t  + 4426686t t  - 248695t t  - 451433t t  + 4023298t t  + 1544133t  + 1234759t t  + 90290t t  - 2545586t t  - 1150739t t  - 1787777t t  + 1802911t t  + 2332905t ,
                             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
                      2129392t  - 2762640t t  + 2068935t  - 665335t t  + 1654813t t  + 3097870t  + 4330772t t  + 4106480t t  - 2058838t t  + 4922805t  + 3532045t t  + 1114886t t  + 1503885t t  - 4151451t t  - 2973500t  - 4218136t t  - 3804423t t  + 2726265t t  + 4523918t t  - 210321t t  + 3766839t  + 1694279t t  + 2490538t t  + 758476t t  + 4575547t t  - 4106063t t  + 1864320t t  + 2128092t ,
                              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
                      - 2140546t  - 4355211t t  - 346596t  - 588652t t  + 4481501t t  + 2091130t  - 1975694t t  + 2412002t t  + 4759999t t  + 2932610t  + 4218913t t  - 229496t t  - 1516819t t  - 2986701t t  - 3352594t  - 4471899t t  - 2629530t t  + 676729t t  - 2338265t t  - 4078308t t  + 3644505t  - 1738017t t  - 3885031t t  - 4740959t t  - 3459143t t  - 1142348t t  - 2678247t t  + 517514t ,
                                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
                      - 2759923t  + 3719767t t  - 1969015t  - 969262t t  - 819291t t  + 3628124t  - 4781195t t  - 4169557t t  - 2707984t t  - 2673395t  - 2779068t t  - 3829406t t  + 2791987t t  + 1229602t t  + 4902870t  + 800952t t  - 3821621t t  - 868567t t  + 984931t t  + 974541t t  - 4293434t  - 1633607t t  + 2030651t t  - 3564414t t  - 4859491t t  + 3153438t t  - 656219t t  + 399948t ,
                                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 :

For the programmer

The object parametrize is a method function.