parametrize -- parametrization of linear varieties and hyperquadrics

Synopsis

• Usage:
parametrize I
• Inputs:
• I, an ideal, the ideal of a linear variety or of a hyperquadric
• Outputs:
• , a birational map phi such that I == image phi

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 :

• "parametrize(Ideal)"
• "parametrize(PolynomialRing)"
• "parametrize(QuotientRing)"

For the programmer

The object parametrize is .