# Singular

#### D.15.22.3 modfWalk

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); ideal J = modfWalk(I); J; ==> J[1]=z4+1/5z2+4/5 ==> J[2]=y5+1/11y3+2/11 ==> J[3]=x3+1/3x2+1/3 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 = modfWalk(I,"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]=b2a6-1275460856846934902527/2619112706933760ba13-9203207900045442436\ 039/4583447237134080ba12-86229451659721876411/6261539941440ba11-63/2ba10-\ 176103105371/907641397248a21-306836641361771/103471119286272a20-556810728\ 2265313/181074458750976a19+463053737062394039071/32593402575175680a18+233\ 493474751858876283/2037087660948480a17+66864105293320847653/2910125229926\ 40a16-2870620980805387/101854383047424a15-9651151354818523/25463595761856\ a14-471264260028168789475/611126298284544a13-489585210177544137565/152781\ 574571136a12-4323891804194718805849/190976968213920a11+4/7a6 ==> J[4]=b3a5+1165198340059361507/60324030330315296b2a8+2834258341382482451/3\ 3932267060802354b2a7-3535202147650393/45243022747736472b2a6-1743263488518\ 66272086637/108100662351925010432ba13-180877903989284763838711/2702516558\ 7981252608ba12-2783949606995368438959121/60806622572957818368ba11+7423924\ 5100658253/30162015165157648ba10+4/7ba5+82576115266283735625891/172961059\ 7630800166912a18+2949131532722419989261/7721475882280357888a17+8257374787\ 8379091193883/108100662351925010432a16-194476798666289267/270251655879812\ 52608a15+10816123814459571297/6756291396995313152a14-13707944038216024518\ 9169/54050331175962505216a13-143935810252288492483523/1351258279399062630\ 4a12-108976370407053801898907/1447776727927567104a11+9/2a10+18a9+11651983\ 40059361507/105567053078051768a8+5668516682764964902/118762934712808239a7\ -3535202147650393/79175289808538826a6 ==> J[5]=b4-63/2b3a4+53590985045967705613474269749/1407238979545595225088b2a8\ +11625419776309240542435260807/58634957481066467712b2a7+12739505342603800\ 3554252983797/175904872443199403136b2a6+4/7b2-761164231440237277536357328\ 493047/700492292040474067599360ba13-104521491576210733019159653207691/250\ 17581858588359557120ba12-11826338279466524349922531923473911/394026914272\ 766663024640ba11+286749359828001390655523145503/39089971654044311808ba10-\ 61261515/8ba9-525086793/32ba8+3969/16ba7-19845/4ba6+9/2ba5+45295478141475\ 95860677850296017/138368847810464013352960a18+247345844501521328455296898\ 6663/9729059611673250938880a17+324025940290811430696988658749687/70049229\ 2040474067599360a16-4286410117811597590771586510563/350246146020237033799\ 68a15+4054710919870705842300540149/2918717883501975281664a14-394300742086\ 35887294045314648189/23349743068015802253312a13-1139723514755969617170188\ 24182045/17512307301011851689984a12-3226716701597602362857429388525683/65\ 671152378794443837440a11+771901337907/64a10-1701/4a9+80528689834593164587\ 767860213/2462668214204791643904a8+11625165236461787969048941667/10261117\ 5591866318496a7+127391998947868572673617154117/307833526775598955488a6 ==> J[6]=da+13042589597338317401292876694323783641/32769687632608435019205956\ 084751360b8-339347956624707128686162624470779591/136540365135868479246691\ 48368646400b7a+44776090034814931064318387941081165483/3276968763260843501\ 9205956084751360b7+72711086789965200830753309360861163/341350912839671198\ 1167287092161600b6a2+17945571338576304647577664032291913/1092322921086947\ 833973531869491712b6a+1005252013115630519751071456639664155699/5734695335\ 70647612836104231483148800b6+9615448034722990303485382435238901/136540365\ 13586847924669148368646400b5a3+663547630395349792235158387078347149/13654\ 036513586847924669148368646400b5a2-10588152408989285294663152330377654/37\ 3352560918390372940172025705175b5a+8954607782583032024373974051595847951/\ 5734695335706476128361042314831488b5-16768783292422657909931339309776827/\ 6827018256793423962334574184323200b4a4+507941899552434600769579126963767/\ 170675456419835599058364354608080b4a3+33005810917176973093954466999277846\ 9/6827018256793423962334574184323200b4a2+64107680452610290564228851734203\ 787/3413509128396711981167287092161600b4a+1376877864134098113390589464572\ 831613/853377282099177995291821773040400b4-643284910890587948236877864337\ 714771/9557825559510793547268403858052480b3a5-130190339942501750272181116\ 13508381/11947281949388491934085504822565600b3a4+137363543353185575764076\ 8919319843/3413509128396711981167287092161600b3a3+66354763039534979223515\ 8387078347149/23894563898776983868171009645131200b3a2-1380791010410641347\ 641955610911797/170675456419835599058364354608080b3a+30455774007493462026\ 8172466495764041/682701825679342396233457418432320b3+57088298465273022117\ 56158284190858113/95578255595107935472684038580524800b2a6+188328905241473\ 4766990421180196597/5973640974694245967042752411282800b2a5-20182594001309\ 76681444622383138147/11947281949388491934085504822565600b2a4+725631285074\ 90657252797018137681/42668864104958899764591088652020b2a3+352799606098503\ 36937362983980868571/1706754564198355990583643546080800b2a2+9160576385145\ 641871627802519827657/1706754564198355990583643546080800b2a+5063824837275\ 183915708779051981113317/11947281949388491934085504822565600b2+2139710638\ 608916733916466261891227/1365403651358684792466914836864640ba8-2207819314\ 4537153891260371881262339/546161460543473916986765934745856ba7-5451366058\ 15586548522667378314077621/3413509128396711981167287092161600ba6-65631655\ 441612907465368619382679107/1706754564198355990583643546080800ba5-189/200\ ba4-13233263602544734530107233305922887/682701825679342396233457418432320\ 0a9-3555188132635967197804852869818541/1365403651358684792466914836864640\ a8+230691784778909201021851072404959103/477891277975539677363420192902624\ 00a7-2239172271256758942944178285566542047/238945638987769838681710096451\ 31200a6+9/5a4 ==> J[7]=db2-52682490423871082296629464283815234/5824767849546495553439269268\ 4816735da+4/7d-37194377146998694799849872861154888794159/6390602440645297\ 9786305125688484646400b8+5562401664632673692771437240347563428391/9319628\ 5592743928855028308295706776000b7a-60969992378608072411136561729066173884\ 2371/319530122032264898931525628442423232000b7-57382636249315287334130146\ 46817239405443/124261714123658571806704411060942368000b6a2-10847266966441\ 319068560579895722895039907/372785142370975715420113233182827104000b6a-27\ 35710626254612029118700402548331033168619/1118355427112927146260339699548\ 481312000b6+180895821889433491876619053925823351/340070372533274690220865\ 93065392000b5a3-13485580971560996606666068965982819343311/186392571185487\ 857710056616591413552000b5a2+3705875600138678704564240261243242140767/543\ 64499929100625165433179839162286000b5a-6096801718704727096493321985237292\ 10634851/279588856778231786565084924887120328000b5+4111550062925787349119\ 083832946689053/1479306120519744902460766798344552000b4a4-329990252397055\ 256446695220829192741/147930612051974490246076679834455200b4a3-5500816426\ 513609089701808086792453204443/72485999905467500220577573118883048000b4a2\ -10849280138379535415092613505685915291543/326186999574603750992599079034\ 973716000b4a-2469122007101831733334304136225298205357/1109479590389808676\ 845575098758414000b4+778576881370029841362906650943590232049/621308570618\ 2928590335220553047118400b3a5+16417021965805756241925867114270711739/1553\ 2714265457321475838051382617796000b3a4+1246566840317431899813433506439313\ 31/59512315193323070788651537864436000b3a3-134855809715609966066660689659\ 82819343311/326186999574603750992599079034973716000b3a2+11337194154659923\ 308002619476290128559/582476784954649555343926926848167350b3a-20322013998\ 4954898395766259918932227475777/326186999574603750992599079034973716000b3\ -17539768590087896897467462966094000442659/144971999810935000441155146237\ 766096000b2a6-12888123469427488410840380363220212653/27182249964550312582\ 716589919581143000b2a5+83962014758738499482259965901772639/77663571327286\ 60737919025691308898000b2a4+135991175566664387828446320649341139/25887857\ 1090955357930634189710296600b2a3-109838629128666311588698057281225716407/\ 3883178566364330368959512845654449000b2a2-3515158182804584308916309399303\ 186117/369826530129936225615191699586138000b2a-31615307392746128515579536\ 862930855381313/54364499929100625165433179839162286000b2+4763451990675051\ 546321234083273260627637/48323999936978333480385048745922032000ba7+157734\ 9394138160739878988144160771568879/5177571421819107158612683794205932000b\ a6+1298061529163168249080815931634187146059/18121499976366875055144393279\ 720762000ba5+70513412412278721745933349799575459/616377550216560376025319\ 49931023000ba4-27/50ba3+3900387310186793261329099028069335191/17258571406\ 06369052870894598068644000a9-3116634130717294137857923622934635379/493102\ 040173248300820255599448184000a8-2437307904588001673427936587902645802933\ /72485999905467500220577573118883048000a7+4675971994304336297474732550069\ 231267517/36242999952733750110288786559441524000a6-5268249042387108229662\ 9464283815234/32359821386369419741329273713787075a4+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+1148200/372848203cb-1524024/27175525ca+4534789049/1391386880d2b2\ a-2324463127/347846720d2ba+35/64d2b+5/96d2a2+5/24d2a+45245274457/10702976\ d2-7/192db5-6353027389387677/12174635200db3a-4497962717/18261952800db3+45\ 245274457/13111145600db2a3+228471973/42611223200db2a2-21815387/1521829400\ db2a-516999/43480840db2+750069/434808400dba4+907911207105139/42611223200d\ ba3+30252540938799/2130561160dba2-1771932852381621/3043658800dba-23707882\ 7/1521829400db+383841663/434808400da5+141/40da4+341903607/173923360da3-63\ 53027418401357/44741784360da2-12706055163515359/22370892180da-73857/10870\ 210d+1728714797635/34784672b6-5208/27175525b5a+6353027467398221/639168348\ 00b5-686156751/869616800b4a2+27783/16723400b4a+6173979975409/217404200b4-\ 69453/760914700b3a3-686156751/434808400b3a2+1321617/108702100b3a+12706043\ 95725157/22370892180b3-14003199/434808400b2a4-4652991/217404200b2a3-98022\ 393/217404200b2a2+3969/4180850b2a-7938/2090425b2-2222633311245/17392336ba\ 5-55565845783569/108702100ba4-9236529/190228675ba3-98022393/108702100ba2+\ 190503/27175525ba-762012/27175525b-571509/108702100a6-19059082255204071/7\ 4569640600a5-38118165490546077/37284820300a4-664713/54351050a3 ==> 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 = modfWalk(I,"Dp",w); J; ==> J[1]=d2 ==> J[2]=c2+3a3+1/3cb2+5/3d ==> J[3]=ca2+4ca+7/2b3+2b ==> J[4]=cda-6/7ba3-2/21cb3-12/7a3-4/21cb2+1/7dba-10/21db-20/21d ==> J[5]=db4a-192/49b2a3-64/147cb4+2db3a-768/49ba3-256/147cb3-4db4+60/49db2a-\ 768/49a3-256/147cb2-8db3+120/49dba-656/147db2-1952/147db-1280/147d ==> J[6]=db6+333576da3-189/4b8+15876b4a2-5186640/2401b2a3+1815960/49cdb2+8395\ 442084/7203cb4-2304/49cb2a-43215/7db5+199982018/49db3a-4664160/7dba2-126b\ 7+72/7b5a+31752b3a2-20742528/2401ba3-1012272/49cdb+16787427752/7203cb3-46\ 08/49cba-129552/49db4+864216/2401db2a-117b6+144/7b4a+9072b2a2-20738496/24\ 01a3-20160cd+4791470576/7203cb2-172796/49db3+5599841736/2401dba-1296/7b5+\ 288/49b3a+18144ba2+9326304/7cb-19529120/7203db2-1188/7b4+576/49b2a-345655\ 04/7203db-3168/49b3-34564160/7203d-3312/49b2 ==> J[7]=cb3a-5/8da3-245/1024b8-7/32b6a+2/343b2a3+5/112cdb2-108013/49392cb4+2\ 5/7cb2a-3/256db5-15431/2016db3a+5/4dba2-245/256b7-25/32b5a+8/343ba3+5/84c\ db-107981/24696cb3+16/7cba-47/2688db4+53/24696db2a-91/256b6-5/8b4a+8/343a\ 3-5/84cd-15403/12348cb2-12/7ca+1/192db3-12015/2744dba+65/32b5-1/14b3a-5/2\ cb-667/98784db2+85/64b4-2/7b2a+61/3087db-3/112b3+3/14ba+40/3087d+93/112b2\ -6/7b ==> J[8]=cb5-4/7db2a2+96/7b2a3-4/441cdb2+116/21cb4-8/7dba2+384/7ba3-8/441cdb+\ 32/3cb3+14db4-16/7db2a+384/7a3+176/21cb2+28db3-32/7dba+16/7cb+328/21db2+9\ 76/21db+640/21d ==> J[9]=cdb3+18da3+2cdb2+63cb4+441/2db3a-36dba2+126cb3+36cb2+126dba+72cb ==> J[10]=ba4+5/72da3+245/9216b8+7/288b6a-2/3087b2a3+2a4-5/1008cdb2+108013/44\ 4528cb4-11/63cb2a+1/768db5+15431/18144db3a-11/36dba2+245/2304b7+25/288b5a\ +12340/3087ba3-5/756cdb+206765/222264cb3-16/63cba+47/24192db4-53/222264db\ 2a+91/2304b6+5/72b4a+24688/3087a3+5/756cd+114187/111132cb2+4/21ca+7055/17\ 28db3+9271/24696dba-65/288b5+1/126b3a+5/18cb+667/889056db2+10/9da-85/576b\ 4+2/63b2a+126506/27783db+1/336b3-1/42ba+123440/27783d-31/336b2+2/21b ==> J[11]=b3a3+4/63db2a2+7203/64b8+52/21b2a3+18526/3969cdb2-32/189cb4-49/48db\ 5-1/6db3a+8/63dba2+7203/16b7-44/21ba3+37052/3969cdb-20/27cb3-259/72db4-5/\ 63db2a+9261/16b6-128/21a3-176/189cb2-113/36db3+32/63dba+1029/2b5-16/63cb-\ 257/378db2+2205/4b4-556/189db+147b3-640/189d+147b2 ==> J[12]=b5a2-4/441db2a2+129724da3-49/16b8+6178b4a2-864464/1029b2a3+302660/2\ 1cdb2+4197721018/9261cb4-128/7cb2a-28823/12db5+99991009/63db3a-114271928/\ 441dba2+49/4b7+4b5a+86468/7b3a2-3457184/1029ba3-506216/63cdb+8393713780/9\ 261cb3-256/7cba-129625/126db4+432008/3087db2a+133/4b6+8b4a+24712/7b2a2-34\ 56512/1029a3-494080/63cd+2395735192/9261cb2-86405/63db3+2799920668/3087db\ a-2b5+16/7b3a+49408/7ba2+518128cb-9766570/9261db2+9b4+32/7b2a-17280544/92\ 61db-36/7b3-17282560/9261d-44/7b2 ==> J[13]=b7a-296432/7da3+109/16b8+2b6a-2016b4a2+4610560/16807b2a3-14521796/3\ 087cdb2-22401156488/151263cb4+8000/343cb2a+1382867/1764db5-1600806884/308\ 7db3a+4148384/49dba2+77/4b7-194/49b5a-4032b3a2+18438656/16807ba3+8103808/\ 3087cdb-44793095440/151263cb3+13568/343cba+230189/686db4-6908840/151263db\ 2a+533/28b6-312/49b4a-1152b2a2+18435072/16807a3+7901200/3087cd-4261620128\ /50421cb2-4864/343ca+1382435/3087db3-4980595264/16807dba+2078/49b5+176/34\ 3b3a-2304ba2-8294976/49cb+52056614/151263db2+2435/49b4-1472/343b2a+921811\ 84/151263db+1900/343b3+608/343ba+30725120/50421d+8044/343b2-2432/343b ==> J[14]=b9+1037536/343da3+4b8+144b4a2-329408/16807b2a3+148256/441cdb2+15987\ 36592/151263cb4-24704/441db5+799730504/21609db3a-296064/49dba2+40/7b7+288\ b3a2-1317376/16807ba3-4047040/21609cdb+3196814624/151263cb3+2048/2401cba-\ 24db4+164704/50421db2a+40/7b6-64/343b4a+576/7b2a2-3840/49a3-3950080/21609\ cd+886720/147cb2+4096/2401ca-98816/3087db3+1036320/49dba+272/49b5-128/343\ b3a+1152/7ba2+592000/49cb-3721504/151263db2+1152/343b4-256/2401b2a-658688\ 0/151263db+1024/343b3-512/2401ba-6400/147d+1920/2401b2+2048/2401b ==> J[15]=dba3-7cb4-49/2db3a+4dba2-14cb3-4cb2-14dba-8cb ==> J[16]=db3a2+1323da3+63b4a2-60/7b2a3+147cdb2+194441/42cb4-49/2db5+64843/4d\ b3a-2646dba2+126b3a2-240/7ba3-82cdb+194401/21cb3-21/2db4+10/7db2a+36b2a2-\ 240/7a3-80cd+55486/21cb2-14db3+64847/7dba+72ba2+5292cb-226/21db2-400/21db\ -400/21d ==> J[17]=a5+4a4-7/6cb3+5/9da2-7/18b5-2/3cb+20/9da-2/9b3 ==> J[18]=da4-2/49b2a3-2/441cb4-1/63db3a-4/49ba3+10/63cdb-4/441cb3+32/7cba+1/\ 147db2a-b4a+20/63cd+64/7ca-2b3a-10/441db2+4b4-4/7b2a-20/441db+8b3-8/7ba+1\ 6/7b2+32/7b ```