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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 0.00545167 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 0.350246 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
                      - 682809t  + 3459943t t  + 4096634t  + 4491457t t  + 3754509t t  + 4876632t  - 1479422t t  - 2107241t t  + 1323384t t  + 2964353t  + 1158570t t  + 1245987t t  + 1835032t t  + 2549634t t  + 4498087t  + 4652720t t  - 1140139t t  - 4465959t t  - 2303619t t  + 159311t t  - 2032648t  - 1312082t t  + 3706671t t  - 2270713t t  - 2442511t t  - 3356670t t  + 1661576t t  + 1564174t ,
                               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
                      4103505t  - 2848088t t  - 4773227t  - 4065202t t  + 529963t t  - 1851540t  + 3093732t t  + 2067167t t  - 1522819t t  - 3511727t  - 2388328t t  + 602140t t  + 1616211t t  - 2532757t t  + 68061t  - 2776090t t  - 3037765t t  - 3402041t t  + 3710646t t  + 4385373t t  + 1825455t  + 522562t t  + 2080620t t  + 4100165t t  + 46583t t  + 4842280t t  - 166760t t  + 1321220t ,
                              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
                      1737729t  + 3893480t t  + 1307753t  - 2803264t t  - 144194t t  - 1593170t  + 1966600t t  + 2984247t t  + 1788998t t  + 1188706t  + 918591t t  + 538658t t  + 354108t t  - 2250928t t  - 2962346t  + 2256168t t  + 2910935t t  - 179477t t  - 3470248t t  - 3185347t t  - 1525181t  + 2199648t t  + 1623395t t  + 2028017t t  - 1425889t t  - 4560981t t  + 762986t t  - 399277t ,
                              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
                      2874317t  + 2341356t t  + 299627t  + 3775841t t  + 3948288t t  - 3661865t  - 654258t t  + 4673879t t  + 3578624t t  + 656392t  + 4614050t t  + 1411342t t  + 4427288t t  + 1862847t t  + 927489t  + 3279171t t  - 846000t t  + 1701222t t  - 2325665t t  - 2638458t t  + 1530319t  - 3692583t t  + 4985662t t  + 1156916t t  - 4835040t t  + 395424t t  - 1518393t t  - 2077777t ,
                              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
                      - 2372546t  + 4523687t t  - 537627t  - 2153189t t  - 4944054t t  - 4499317t  + 1926472t t  + 4373901t t  + 4833787t t  + 27827t  - 4046783t t  - 3544928t t  - 3089335t t  + 200924t t  - 515082t  + 1139327t t  - 3185695t t  + 79197t t  - 2933537t t  + 4597637t t  + 2296896t  - 262732t t  - 1972027t t  + 4669800t t  - 295236t t  + 3395394t t  - 1212151t t  - 4045281t ,
                                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
                      1297307t  - 4801047t t  + 610262t  + 4331721t t  - 943782t t  + 903568t  - 1526490t t  - 3237988t t  - 1435517t t  + 90540t  + 2563521t t  + 2822792t t  - 4972213t t  + 881578t t  - 3112646t  - 1054213t t  - 3480765t t  - 1869591t t  - 3200205t t  + 3372963t t  + 1947054t  + 486661t t  - 1044409t t  - 2517569t t  - 1440712t t  + 2174453t t  + 3317362t t  - 4766209t ,
                              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
                      567434t  + 1473334t t  + 4978354t  - 1068566t t  + 4315634t t  - 1183092t  - 988513t t  - 3860316t t  - 3210583t t  + 2720158t  + 4127604t t  - 2685021t t  - 593322t t  - 1061777t t  + 4111419t  - 4322904t t  + 2678437t t  - 538473t t  + 3691782t t  + 3595267t t  + 3965155t  + 698714t t  + 719273t t  + 439669t t  - 3657140t t  + 293622t t  + 2430882t t  + 4463926t ,
                             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
                      2709382t  + 3971019t t  + 2001285t  - 4322393t t  - 4546445t t  + 3859985t  - 2090474t t  - 1192533t t  - 655282t t  + 3432959t  - 2812564t t  - 2390207t t  - 3164797t t  - 823959t t  - 1217738t  + 3230081t t  + 4397155t t  - 871527t t  + 1380223t t  - 1612670t t  - 3261705t  - 3629885t t  + 348918t t  - 2313683t t  + 3828352t t  + 2103388t t  - 1907739t t  - 3790233t ,
                              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
                      - 4267374t  - 2054631t t  + 1778625t  - 3360715t t  - 1607780t t  - 4858102t  - 4672480t t  - 1400351t t  - 3300435t t  + 4427649t  + 4546238t t  - 996600t t  - 275531t t  - 3653171t t  + 3734431t  + 1214211t t  + 3776085t t  + 3540077t t  + 2050999t t  - 2757378t t  + 4100511t  - 4184559t t  + 2524520t t  - 4185103t t  - 2179409t t  - 4949382t t  + 4531450t t  - 2452562t ,
                                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
                      3836175t  + 1907460t t  - 4913382t  + 3400588t t  - 1705892t t  - 2358846t  - 1482583t t  - 2731926t t  + 3731588t t  + 3633724t  + 2150988t t  + 672173t t  - 1395357t t  - 3453452t t  - 2177108t  + 1190285t t  - 2824039t t  - 3611723t t  - 934833t t  - 3581702t t  - 2933478t  - 3416967t t  + 3431989t t  - 3941840t t  + 889997t t  - 43439t t  + 219075t t  + 4026790t
                              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 :