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D.15.9.2 modrWalk

Procedure from library modwalk.lib (see modwalk_lib).

Return:
a standard basis of I

Note:
The procedure computes a standard basis of I (over the rational numbers) by using modular methods.

Example:
 
LIB "modwalk.lib";
ring R1 = 0, (x,y,z,t), dp;
ideal I = 3x3+x2+1, 11y5+y3+2, 5z4+z2+4;
I = std(I);
ring R2 = 0, (x,y,z,t), lp;
ideal I = fetch(R1, I);
int radius = 2;
ideal J = modrWalk(I,radius);
J;
==> J[1]=x3+1/3x2+1/3
==> J[2]=z4+1/5z2+4/5
==> J[3]=y5+1/11y3+2/11
ring S1 = 0, (a,b,c,d), Dp;
ideal I = 5b2, ac2+9d3+3a2+5b, 2a2c+7abd+bcd+4a2, 2ad2+6b2d+7c3+8ad+4c;
I = std(I);
ring S2 = 0, (c,d,b,a), lp;
ideal I = fetch(S1,I);
// I is assumed to be a Dp-Groebner basis.
// We compute a lp-Groebner basis.
ideal J = modrWalk(I,radius,"Dp");
J;
==> J[1]=a25+16a24+96a23+256a22+256a21+256/9a20+1024/3a19+2048a18+65536/9a17+\
   32768/3a16+16384/81a15+131072/81a14+1048576/81a13+1048576/27a12+1048576/9\
   a11
==> J[2]=ba11+1522867351997104938459/91668001658017308797687087104a24+4293036\
   9782248629690765/91668001658017308797687087104a23+80925218629630777478637\
   /22917000414504327199421771776a22+7108535670237178684767/2864625051813040\
   899927721472a21-3255817194541612658349/89519532869157528122741296a20+5380\
   8965391546362724459/358078131476630112490965184a19+1534729815590907963215\
   01/358078131476630112490965184a18-260815719913165309506063/44759766434578\
   764061370648a17-1485276141860757031491027/89519532869157528122741296a16-4\
   92332725360316960775/22379883217289382030685324a15+7423992361030571232440\
   /16784912412967036523013993a14-17640364913371983121693/167849124129670365\
   23013993a13-37723213977586186442564/5594970804322345507671331a12+92047580\
   41857159721472414/5594970804322345507671331a11
==> J[3]=b2a6-63/2ba10+41087306587333357057895823883/924013456712814472680685\
   83800832a24+93915562116924232413944264677/1320019223875449246686694054297\
   6a23+1314662746341964624103002499857/30800448557093815756022861266944a22+\
   2628268795042931967685617407557/23100336417820361817017145950208a21+32861\
   7969148352577032114618159/2887542052227545227127143243776a20+175631724829\
   284757906915538269/12993939235023953522072144596992a19+303812212296039043\
   29090032467/206253003730538944794795945984a18+581332950565269518541458030\
   9/6445406366579342024837373312a17+1392577308804410648719627124495/4060606\
   01094498547564754518656a16+728489619081836101063651608791/135353533698166\
   182521584839552a15+80182830319998353431517995037/913636352462621732020697\
   666976a14+22597043807001043905240513127/32629869730807919000739202392a13+\
   2699689128255025271541196980091/456818176231310866010348833488a12+6291773\
   289016174025274735397/352483160672307766983293853a11+4/7a6
==> J[4]=b3a5+4/7ba5-5398327059462849163101023479/739210765370251578144548670\
   406656a24+3500067651845908053488406611/92401345671281447268068583800832a2\
   3+81936880336538263803253409291/46200672835640723634034291900416a22+85587\
   506267677081700930548967/6600096119377246233433470271488a21+4349147060430\
   84846670081811347/11550168208910180908508572975104a20+1021713731964491721\
   016405426709/25987878470047907044144289193984a19+189087309367688338214840\
   3875/812121202188997095129509037312a18+3873208423822126582196454031/10312\
   6501865269472397397972992a17+104243440049859097996976327663/4060606010944\
   98547564754518656a16+52530075262606982469983545909/5075757513681231844559\
   4314832a15+6154556265260978917193662164647/365454540985048692808279066790\
   4a14+138400848395446486358821423/5639730570756924271732701648a13+68577767\
   53456717192310999393/38068181352609238834195736124a12+3439955547980759942\
   57860510037/228409088115655433005174416744a11+9/2a10+18a9
==> J[5]=b4-63/2b3a4+4/7b2+3859043113737/128ba10-61261515/8ba9-525086793/32ba\
   8+3969/16ba7-19845/4ba6+9/2ba5-317530772199391516703685862633925319890715\
   5/9853890705461273608546303647569412096a24-889000025950798002413919214249\
   17407047623059/17244308734557228814956031383246471168a23-4443358152931308\
   1841158459544460024461570519/1437025727879769067913002615270539264a22-443\
   94201884268758058296712245149507940575423/5388846479549134004673759807264\
   52224a21-88794706503699382133468723144783169844707611/1077769295909826800\
   934751961452904448a20-1763218978464215084468461907410740543335425/1732129\
   22556936450150227993804931072a19-4436403904696107325808495191699219229446\
   85/4210036312147760941151374849425408a18-21791665871565292166347154127869\
   734967876143/33680290497182087529210998795403264a17-382244791239220889094\
   021792022832016196677083/151561307237319393881449494579314688a16-25381669\
   887869257097552146559952295821945667/6315054468221641411727062274138112a1\
   5-3081988985408645474889243822837753592680251/487161344691383766047516232\
   57636864a14-42991086686519981772739427292780808208480959/8525323532099215\
   9058315340700864512a13-44791324614077928834488691100822888689474979/10656\
   654415124019882289417587608064a12-250969820105414481651193596059252574282\
   0133/197345452131926294116470696066816a11+771901337907/64a10-1701/4a9+437\
   53689/4a8-19845/8a7-19845/2a6
==> J[6]=da-1323/800b3a4+63/20b2a4+2701701074331/1280ba10-61269453/320ba9+306\
   291699/160ba8-27783/16ba6-189/200ba4-685883099436497069901143509465258298\
   0621641/123173633818265920106828795594617651200a24-5486790468650205133145\
   8584531975294116209709/61586816909132960053414397797308825600a23-60954530\
   98260341081175657932145681344303369/1140496609428388149137303662913126400\
   a22-54839253809480536502430419967744127237862429/384917605682081000333839\
   9862331801600a21-6855741428155530311932580353074238311128479/481147007102\
   601250417299982791475200a20-2878029541644786739384025698911163129301521/1\
   732129225569364501502279938049310720a19-539637460347386518746735121876349\
   5466944867/288688204261560750250379989674885120a18-1360851075145362976080\
   2370233473935440353147/120286751775650312604324995697868800a17-2261317453\
   3404843047583509526614944559001629/54129038299042640671946248064040960a16\
   -19899168362053216404911620136404443756991/309379505595808417192193918976\
   00a15-3443170203569948104100043669593479708510001/30447584043211485377969\
   7645360230400a14-27203224432640761938740031635681832222207391/30447584043\
   2114853779697645360230400a13-11007706754707314979975768772869745826795967\
   3/152237920216057426889848822680115200a12-1380727908059280736173048219943\
   2331439121223/6343246675669059453743700945004800a11-1929592445193/640a10+\
   437570343/1600a9+164090367/100a8+3969/8a6+9/5a4
==> J[7]=db2+4/7d+81/160b3a4-189/200b3a3+9/5b2a3+250047/32ba10+4750893/320ba9\
   -567/40ba8-567/10ba7+81/280ba4-27/50ba3-627430952592078879073202920720376\
   153131/5173292620367168644486809414973941350400a24-2906859466655459274817\
   4299538370243493/18476045072739888016024319339192647680a23-69048240138740\
   571289193103124930690109/11975214398998075565941688460587827200a22+488085\
   08639174871490992992380443024047/11547528170462430010015199586995404800a2\
   1+5093069077149293873574182379116672811143/808326971932370100701063971089\
   67833600a20+3286117544936595300844164893345508176887/36374713736956654531\
   547878699035525120a19-57124126734867585149642894515011901057/189451634046\
   6492423518118682241433600a18-296407013643924816106633846582502767991/2526\
   021787288656564690824909655244800a17-193831832299525900077669479610197531\
   4379/11367098042798954541108712093448601600a16+49045157132492520042165362\
   0398209599/433131307834131784069071486566400a15+9565278519988924726047202\
   0815175898220317/25575970596297647717494602210259353600a14-51800242230268\
   556716933919846651112061/456713760648172280669546468040345600a13-15498981\
   08498974500694541229046869490537/1598498162268602982343412638141209600a12\
   -700479417253540849801700529315888259/26641636037810049705723543969020160\
   a11+20611017/1600a10+21504771/320a9+243/50a8-1701/100a7+36/35a3
==> J[8]=d2
==> J[9]=ca-80796665/1042771968d2b2a+80796665/521385984d2ba-456425375/6069043\
   2d2a-3048690575/30345216d2+91285075/182071296db4a+35/9216db4-1088785/5689\
   728db3a2+2258752616171/182071296db3a+24027/2528768db3-8162999/79656192db2\
   a3-6539063/79656192db2a2+25495277/45517824db2a+5095/1264384db2-5243/15804\
   8dba4-753114969703/1486915584dba3-753467060575/2230373376dba2+30870844232\
   377/2230373376dba+24027/4425344db-7/256da5+5/96da3+342228711385/101380608\
   da2+342235311685/25345152da+2645/2489256d-68441518085/57931776b6+8617/189\
   6576b5a-752908955671/318624768b5+7/256b4a2-2163/79024b4a-752734601911/111\
   5186688b4+529/316096b3a3+7/128b3a2-12811/59268b3a-752421061495/557593344b\
   3+311787/632192b2a4+529/158048b2a3+1/64b2a2-309/19756b2a+309/4939b2+68441\
   518085/22529024ba5+752978795959/61954816ba4+529/553168ba3+1/32ba2-1/8ba+1\
   /2b+3/32a6+68445742277/11264512a5+68447062337/2816128a4+529/276584a3
==> J[10]=cb-1068739/27586560d2b2a+1068739/13793280d2ba-76214943/20230144d2a-\
   508115055/10115072d2+25404981/101150720db4a-7203/1024db4+453789/790240db3\
   a2+125486265849/20230144db3a-25399101/12643840db3-324051/6321920db2a3+194\
   523/790240db2a2+11921259/25287680db2a-9921275/1264384db2-27783/632192dba4\
   -12813612147/50575360dba3-12806449187/75863040dba2+104952523865/15172608d\
   ba-31286843/3160960db+1163396585/689664da2+1163396585/172416da-172823/790\
   24d-1628855795/2758656b6+64827/3160960b5a-89587760437/75863040b5+64827/15\
   80480b4a-12804376067/37931520b4+1323/1580480b3a3+9261/790240b3a-127982514\
   91/18965760b3-583443/395120b2a4-10887849/1580480b2a3+9261/395120b2a-9261/\
   98780b2+698081055/459776ba5+38375787849/6321920ba4-6222951/395120ba3+6980\
   37951/229888a5+698037951/57472a4-1555407/395120a3
==> J[11]=cd+3281/1008d2b2a-53819/8064d2ba+35/64d2b+5/96d2a2-155/768d2a+33775\
   /8d2-7/192db5+7/256db4a+7/320db4-1/80db3a2-133413949/256db3a-23/96db3+441\
   /128db2a3+1/64db2a+1/80db2+667316711/31360dba3+667069751/47040dba2-273498\
   58321/47040dba-1/8db+141/160da5+141/40da4+63/32da3-667069751/4704da2-6670\
   69751/1176da+667069751/13440b6+667069751/6720b5-63/80b4a2+667069751/23520\
   b4-63/40b3a2+667069751/11760b3-9/20b2a2-2001209253/15680ba5-2001209253/39\
   20ba4-9/10ba2-2001209253/7840a5-2001209253/1960a4
==> J[12]=c2+1/3cb2+5/3d+3a3
intvec w = 3,2,1,2;
ring S3 = 0, (c,d,b,a), (a(w),lp);
ideal I = fetch(S1,I);
// I is assumed to be a Dp-Groebner basis.
// We compute a (a(w),lp)-Groebner basis.
ideal J = modrWalk(I,radius,"Dp",w);
J;
==> J[1]=d2
==> J[2]=c2+3a3+1/3cb2+5/3d
==> J[3]=ca2+4ca+7/2b3+2b
==> J[4]=cda-6/7ba3+4/7c2-2/21cb3+1/7dba-10/21db
==> J[5]=db4a-192/49b2a3+128/49c2b-64/7cda-64/147cb4+4/63d2ba+2db3a-4db4+60/4\
   9db2a-8db3+8/7dba-656/147db2-32/7db
==> J[6]=db6+333576da3-189/4b8+15876b4a2-5186640/2401b2a3-726526982424497/741\
   1887c2b+88375033207344527935/11206773144cd2+1815960/49cdb2-72655381947596\
   9/2117682cda+8395442084/7203cb4-2304/49cb2a-605450868912/343ca2+3806167/6\
   174d2b3+17676715408101464027/11206773144d2ba-43215/7db5+199982018/49db3a-\
   4664160/7dba2-126b7+72/7b5a+31752b3a2+37367280/16807ba3-1453086299039554/\
   7411887c2-1012272/49cdb+39190949752/16807cb3-4608/49cba-3806167/21609d2b2\
   -129552/49db4+864216/2401db2a-117b6+144/7b4a+9072b2a2-20160cd+4663152/7cb\
   2-2421803475648/343ca-13992486044/50421d2b-172796/49db3-691980396597905/1\
   4823774dba-1296/7b5+288/49b3a+18144ba2+9326304/7cb+185320d2-19529120/7203\
   db2-1188/7b4+576/49b2a+20772144/16807db-302725437624/49b3-3312/49b2-12109\
   01737824/343b
==> J[7]=cb3a-5/8da3-245/1024b8-7/32b6a+2/343b2a3-3632987219924149/7318708992\
   c2b+441947852157122517275/11065887995904cd2+5/112cdb2-3633014627975365/20\
   91059712cda-108013/49392cb4+25/7cb2a-7007533245/784ca2+12978355/6096384d2\
   b3+88398121169095840015/11065887995904d2ba-3/256db5-15431/2016db3a+5/4dba\
   2-245/256b7-25/32b5a+1297475/115248ba3-3633014656425157/3659354496c2+5/84\
   cdb-3240415/1037232cb3+16/7cba-12978355/21337344d2b2-47/2688db4+53/24696d\
   b2a-91/256b6-5/8b4a-5/84cd-5/4cb2-7007533581/196ca-52493006345/37340352d2\
   b+1/192db3-3633078720022405/14637417984dba+65/32b5-1/14b3a-5/2cb-25/72d2-\
   667/98784db2+85/64b4-2/7b2a+6494431/1037232db-7007533251/224b3+3/14ba+93/\
   112b2-7007533581/392b
==> J[8]=cb5-4/7db2a2+96/7b2a3-64/7c2b-4/441cdb2+32cda+116/21cb4-8/7dba2-8/44\
   1cdb+32/7cb3+14db4-16/7db2a+16/7cb2+28db3+16/7cb+328/21db2+16db
==> J[9]=cdb3+18da3+2cdb2+63cb4+441/2db3a-36dba2+126cb3+36cb2+126dba+72cb+10d\
   2
==> J[10]=ba4-2/3c2a+1/9cb3a+14/3cda-1/6dba2+49/12db3+5/9dba+7/3db
==> J[11]=b3a3-2116653/4c2d-2/3c2b2+3171721/72cdb3-12005/2cdba+540241/144cb5-\
   105cba2-18914/81d2b4+2803530031/9072d2b2a+343/144db6+3781687/288db4a-8575\
   /4db2a2+151263/4b4a2+172862/9c2b-7007690605/2268cd2-455/12cdb2+496972/9cd\
   a+44536835/8cb4-112cb2a+24ca2+37828/81d2b3-5605816001/9072d2ba-117649/8db\
   5+467064125/24db3a-3177258dba2+2401/16b7+49/2b5a+151263/2b3a2+345724/9c2-\
   886795/18cdb+44485385/4cb3-644cba-10808/81d2b2-453691/72db4+60025/8db2a+2\
   401/8b6+49b4a+21609b2a2-48020cd+6362405/2cb2+96ca+21616/81d2b-151253/18db\
   3+200318507/18dba+147/2b5+14b3a+43218ba2+6353830cb-7203/2db2-441/2b4+28b2\
   a+77b3-224b2+48b
==> J[12]=b5a2-71978c2d+161746/27cdb3-40062/49cdba+7007/12cb5-100/7cba2-10802\
   9/3402d2b4+400654481954/9529569d2b2a-7/108db6+3090359/1512db4a-147151/441\
   db2a2+5149b4a2+131697064/50421c2b-8011792558855/19059138cd2-407/63cdb2+16\
   2224632/21609cda+2272634/3cb4-320/21cb2a-928/49ca2+108029/1701d2b3-320439\
   1253461/38118276d2ba-18016/9db5+667206751/252db3a-190627550/441dba2+245/1\
   2b7+10/3b5a+72062/7b3a2+263394128/50421c2-1267582/189cdb+4539262/3cb3-184\
   0/21cba-216058/11907d2b2-324371/378db4+1030061/882db2a+245/6b6+20/3b4a+20\
   596/7b2a2-411760/63cd+1298648/3cb2-3712/49ca+432116/11907d2b-24032/21db3+\
   229021214453/151263dba+10b5+40/21b3a+41176/7ba2+2593292/3cb-216238/441db2\
   -30b4+80/21b2a-128/441db-1412/21b3-640/21b2-1856/49b
==> J[13]=b7a-982510/49c2d+152/3087c2b2-26750/583443cd2b+690371/294cdb3-41637\
   5/3087cdba+1555/8cb5-352/49cb3a-1770/343cba2-45548/3969d2b4+95714921047/8\
   168202d2b2a+13/72db6+33602059/49392db4a-1555/14db2a2+25/7b6a+1701/2b4a2+2\
   2131028/21609c2b-637699858825/5445468cd2-13205/6174cdb2+3410320/1029cda+2\
   955327/14cb4-3712/343cb2a+3460272/2401ca2+91096/3969d2b3-255052311793/108\
   90936d2ba-583267/1764db5+18213920215/24696db3a-5900147/49dba2+27/8b7+172/\
   49b5a+1701b3a2+14752600/7203c2-1140997/1029cdb+2950662/7cb3-4968/343cba-1\
   82192/27783d2b2-249961/1764db4+4800685/12348db2a+13/28b6+90/49b4a+486b2a2\
   -3334160/3087cd+5910654/49cb2+13836608/2401ca+370684/27783d2b-583267/3087\
   db3+6078238601/14406dba-275/49b5+716/343b3a+972ba2+11799538/49cb-83381/10\
   29db2-419/49b4-40/343b2a+1728098/343b3+80/343ba-1728/343b2+6918304/2401b
==> J[14]=b9-1035968/1029c2d+2906389690115780/1400846643c2b2-8838957040922773\
   8655/529520031054cd2b-16/63cdb3+1453201255066978/200120949cdba+100/7cb5+8\
   9696424576/2401cba2-5710126/583443d2b4-17679314984106853859/529520031054d\
   2b2a+50db4a-400/49db2a2+4b8+144b4a2-16607680/352947b2a3+5812823384608648/\
   1400846643c2b-56061528920/9529569cd2-16/343cdb2-328192/7203cda+3369446955\
   2/3176523cb4-22126080/16807ca2+17645660/4084101d2b3-11209873228/9529569d2\
   ba-24704/441db5+800681300/21609db3a-296416/49dba2+40/7b7+288b3a2+4352/504\
   21c2-1350880/7203cdb+1037632/49cb3+358785700352/2401cba+167963094352/2858\
   8707d2b2-24db4+1453245855228010/1400846643db2a+40/7b6-64/343b4a+576/7b2a2\
   -3950080/21609cd+297152/49cb2-88475648/16807ca+157312/194481d2b-98816/308\
   7db3+1067312816/50421dba+272/49b5-128/343b3a+1152/7ba2+12096cb-126602144/\
   3176523db2+44848213440/343b4-256/2401b2a-11055872/2401b3-512/2401ba+17939\
   2851072/2401b2-44237824/16807b
==> J[15]=dba3-2/3c2d+1/9cdb3-1/6d2ba+5/9d2b
==> J[16]=db3a2-147/2cdb3-10cdba+63b4a2-32c2b-132cda-576ca2-49/2db5+4db3a+126\
   b3a2-64c2-82cdb-21/2db4+36b2a2-80cd-2304ca-14db3-16dba+72ba2-6db2-2016b3-\
   1152b
==> J[17]=a5+1/9cb2a2-4/3c2a-7/6cb3+5/9da2-2/3cb
==> J[18]=da4-4/21c3+11/63cdba+8/63c2b-1/63db3a+10/63cdb+32/7cba+5/9d2a+2/63d\
   b2a-b4a+4b4-4/7b2a+16/7b2
See also: modular.


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