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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 0.00613374 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 0.391913 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
                      3810457t  - 4062914t t  + 1116613t  - 851076t t  - 2195930t t  + 3328417t  + 2808775t t  + 3336380t t  + 3479811t t  + 509663t  - 2766083t t  - 3444690t t  + 2726555t t  - 2557473t t  - 4863909t  + 851754t t  - 927798t t  - 3350896t t  + 194688t t  - 2808033t t  + 1342608t  - 65040t t  + 3674385t t  - 2921914t t  + 836688t t  + 2019408t t  - 2550783t t  - 2358913t ,
                              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
                      4992910t  + 185995t t  - 2371306t  - 768515t t  + 3969639t t  + 3882955t  + 3479921t t  - 4071403t t  + 156992t t  + 3804571t  + 1683117t t  + 1345651t t  + 1549247t t  - 2829697t t  - 295121t  + 1424766t t  + 2361539t t  + 2739585t t  + 4853729t t  + 3046102t t  - 4335217t  - 1223298t t  - 582618t t  + 3646552t t  + 1813559t t  - 1488018t t  + 4877251t t  + 1686152t ,
                              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
                      4272529t  + 4672518t t  - 2161508t  + 1903854t t  + 3350873t t  + 4848756t  + 2326909t t  + 3806520t t  - 2977201t t  + 1454465t  - 4152392t t  - 2626580t t  - 1010785t t  - 2923446t t  + 2583224t  + 55633t t  - 2130779t t  + 3503436t t  + 3728444t t  - 1700286t t  + 4952699t  + 2589800t t  + 3973889t t  + 3179164t t  - 4107156t t  - 2843390t t  - 16813t t  - 1085942t ,
                              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
                      - 3411911t  + 4356256t t  - 4080528t  - 928208t t  - 4694109t t  + 1327801t  - 2959248t t  - 1549219t t  - 189335t t  + 4540257t  - 4774360t t  - 58071t t  + 1313875t t  + 87746t t  + 4432761t  + 1529459t t  - 2239555t t  + 2498909t t  + 452870t t  - 294162t t  + 4606010t  - 3390523t t  + 3134454t t  + 4528108t t  - 2606448t t  + 2119120t t  - 4154462t t  + 32344t ,
                                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
                      - 2308806t  + 205727t t  + 1230354t  + 3124734t t  - 993233t t  + 2121544t  + 990525t t  - 1732071t t  - 95233t t  + 4005876t  + 5387t t  + 2539085t t  + 2881603t t  + 4218756t t  + 3883261t  - 3805160t t  - 4257560t t  + 2418423t t  - 4155078t t  + 2347319t t  + 117982t  + 1191298t t  - 309794t t  + 2131218t t  + 3849989t t  - 1269579t t  + 3296180t t  + 3248891t ,
                                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
                      - 3256308t  - 2252482t t  - 2124576t  + 1997780t t  + 2589170t t  - 4026572t  - 889774t t  + 3635973t t  + 4819128t t  - 4622666t  + 2289171t t  + 4520710t t  - 1360547t t  + 3385108t t  + 963725t  + 2072517t t  + 1422823t t  + 4342159t t  - 329681t t  + 4134846t t  - 4243472t  + 4553499t t  - 4672822t t  + 3750819t t  + 4005284t t  + 4010049t t  + 1538026t t  - 2676276t ,
                                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
                      - 679742t  + 1681922t t  - 833288t  + 633807t t  - 3661008t t  + 1328095t  + 623025t t  + 3159301t t  + 4403961t t  - 3764145t  + 4308759t t  - 3912479t t  + 1943290t t  - 1445564t t  + 3836197t  + 856111t t  - 4971256t t  + 4302507t t  - 3370199t t  + 909570t t  + 4211631t  - 4403133t t  - 2999917t t  + 164864t t  - 4921165t t  - 2321529t t  - 186745t t  - 3639087t ,
                               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
                      590428t  - 1823110t t  - 3700910t  - 1892560t t  + 3849888t t  + 1246977t  - 579913t t  + 550043t t  - 3820821t t  - 3303851t  - 3294831t t  + 3528513t t  - 2749998t t  + 1659688t t  + 2692588t  + 3074251t t  + 90453t t  + 393626t t  + 3268947t t  + 3477949t t  - 2072315t  - 2060503t t  - 514816t t  - 3026326t t  - 2256663t t  - 4176322t t  - 1842815t t  - 4070214t ,
                             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
                      4919161t  + 3453790t t  - 4531279t  + 875390t t  - 4523064t t  + 1687013t  - 4733481t t  - 4811003t t  - 810756t t  + 502692t  - 1538014t t  + 2406716t t  + 2420012t t  + 2931529t t  + 319119t  + 1346947t t  + 1504285t t  + 571506t t  + 1834909t t  + 4786863t t  + 4008310t  + 4319679t t  - 4627633t t  - 804179t t  - 3369396t t  + 93319t t  + 3289390t t  + 2351630t ,
                              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
                      - 4625543t  - 711145t t  + 3751394t  + 4614781t t  + 1832668t t  + 1434779t  - 2264440t t  - 2071821t t  - 3716203t t  + 2647138t  + 1757911t t  + 2880453t t  - 69019t t  - 4489801t t  + 1578666t  + 638592t t  + 3776258t t  + 3195986t t  + 4523223t t  + 1789506t t  + 2872005t  + 4500079t t  - 1156350t t  + 3132766t t  - 3615163t t  - 1387166t t  + 4612979t t  + 3768408t
                                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 :