# Singular

#### D.15.16.2 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+91259683/91035648db3a+24027/2528768db3-8162999/79656192db2a3-653\ 9063/79656192db2a2+25495277/45517824db2a+5095/1264384db2-5243/158048dba4-\ 109809/632192dba3-173005/632192dba2+265684003/159312384dba+24027/4425344d\ b-7/256da5+5/96da3+5/24da2+35/32da+2645/2489256d+8617/1896576b5a-25921/15\ 8048b5+7/256b4a2-2163/79024b4a+2163/19756b4+529/316096b3a3+7/128b3a2-1281\ 1/59268b3a+15435/19756b3+311787/632192b2a4+529/158048b2a3+1/64b2a2-309/19\ 756b2a+309/4939b2+19467/9878ba4+529/553168ba3+1/32ba2-1/8ba+1/2b+3/32a6+3\ /8a5+63/32a4+529/276584a3 ==> J[10]=cb-1068739/27586560d2b2a+1068739/13793280d2ba-76214943/20230144d2a-\ 508115055/10115072d2+25404981/101150720db4a-7203/1024db4+453789/790240db3\ a2+32184817/10115072db3a-25399101/12643840db3-324051/6321920db2a3+194523/\ 790240db2a2+11921259/25287680db2a-9921275/1264384db2-27783/632192dba4-966\ 217/3160960dba3-43211/395120dba2+1278655/2528768dba-31286843/3160960db-5/\ 48da2-5/12da-172823/79024d+64827/3160960b5a-1351/148170b5+64827/1580480b4\ a-64827/395120b4+1323/1580480b3a3+9261/790240b3a-386/74085b3-583443/39512\ 0b2a4-10887849/1580480b2a3+9261/395120b2a-9261/98780b2-583443/197560ba4-6\ 222951/395120ba3-3/16a5-3/4a4-1555407/395120a3 ==> J[11]=cd+3281/1008d2b2a-53819/8064d2ba+35/64d2b+5/96d2a2-155/768d2a+33775\ /8d2-7/192db5+7/256db4a+7/320db4-1/80db3a2+3/640db3a-23/96db3+441/128db2a\ 3+1/64db2a+1/80db2+63/8dba3+1/32dba-1/8db+141/160da5+141/40da4+63/32da3-6\ 3/80b4a2-63/40b3a2-9/20b2a2-9/10ba2 ==> 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+15876b4a2-5186640/2401b2a3-649481368492976/741\ 1887c2b+105256176776539512385/14942364192cd2+1815960/49cdb2-3247541027722\ 24/1058841cda+8395442084/7203cb4-2304/49cb2a-3785931343620/2401ca2-2038/2\ 1d2b3+21053269526043185513/14942364192d2ba-43215/7db5+199982018/49db3a-46\ 64160/7dba2-126b7+72/7b5a+31752b3a2+37367280/16807ba3-1298995071176512/74\ 11887c2-1012272/49cdb+39190949752/16807cb3-4608/49cba+4076/147d2b2-129552\ /49db4+864216/2401db2a-117b6+144/7b4a+9072b2a2-20160cd+4663152/7cb2-15143\ 725374480/2401ca-525626613437/2117682d2b-172796/49db3-307467391333192/741\ 1887dba-1296/7b5+288/49b3a+18144ba2+9326304/7cb+185320d2-19529120/7203db2\ -1188/7b4+576/49b2a+20772144/16807db-1892965693986/343b3-3312/49b2-757186\ 2687240/2401b ==> J[7]=cb3a-5/8da3-245/1024b8-7/32b6a+2/343b2a3-202984966037233/457419312c2\ b+526377752591243536325/14754517327872cd2+5/112cdb2-101493339520217/65345\ 616cda-108013/49392cb4+25/7cb2a-525872201885/65856ca2-46165/31104d2b3+105\ 285730577936514925/14754517327872d2ba-3/256db5-15431/2016db3a+5/4dba2-245\ /256b7-25/32b5a+1297475/115248ba3-101493340409273/114354828c2+5/84cdb-324\ 0415/1037232cb3+16/7cba+6595/15552d2b2-47/2688db4+53/24696db2a-91/256b6-5\ /8b4a-5/84cd-5/4cb2-525872230109/16464ca-2629319353265/2091059712d2b+1/19\ 2db3-101495342396687/457419312dba+65/32b5-1/14b3a-5/2cb-25/72d2-667/98784\ db2+85/64b4-2/7b2a+6494431/1037232db-525872202389/18816b3+3/14ba+93/112b2\ -525872230109/32928b ==> 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+2744/5c2d+382/15c2b2-1127/15cdb3+693167/15cdba+129659/45cb5+23\ 52/5cb3a-8647/27d2b2a-49/15db4a-8232/5db2a2-1029/10b6a+616/15c2b+86470/27\ cd2+161/5cdb2+1382932/15cda+28812/5cb4+1680cb2a+686/5d2ba-98/15db5-121045\ /6db3a-735/2b5a-112/5c2-224/15cdb+8232/5cb3+5376/5cba+201698/5db4+38/3db2\ a+2058/5b6-294b4a-28cd+16464/5cb2-4032/5ca+3631597/45db3-173024/15dba+147\ 0b5-168/5b3a+115256/5db2+1176b4-672/5b2a+138352/3db+672/5b3+504/5ba+2688/\ 5b2-2016/5b ==> J[12]=b5a2+266/3c2d+32/9c2b2-113/9cdb3+1097044/147cdba+931/2cb5+64cb3a+1/\ 81d2b4-19559/378d2b2a-7/18db6-4/9db4a-117310/441db2a2-14b6a+4b4a2+32/7c2b\ +97795/189cd2+194/63cdb2+6567368/441cda+931cb4+1600/7cb2a-128/7ca2-2/81d2\ b3+133/6d2ba-11/6db5-136889/42db3a-8/441dba2-50b5a+32/7b3a2-320/63c2-124/\ 21cdb+266cb3+1024/7cba+4/567d2b2+410576/63db4+248/147db2a+56b6-40b4a+16/7\ b2a2-400/63cd+532cb2-1280/7ca-8/567d2b+117340/9db3-91334/49dba+200b5-32/7\ b3a+16/7ba2+1642360/441db2+160b4-128/7b2a+3285784/441db-320/7b3+96/7ba+51\ 2/7b2-640/7b ==> J[13]=b7a+2931231/490c2d+1024/735c2b2-864700/27783cd2b+1583507/8820cdb3+2\ 9189312/12005cdba+758/5cb5+3904/245cb3a-5432499127/1555848d2b2a+4/63db6+5\ 37472/15435db4a-3032/35db2a2-52/35b6a-1008b4a2+1266752/15435c2b+905070254\ 5/259308cd2+20176/15435cdb2+554322208/108045cda-8711357/140cb4+26496/343c\ b2a+1440ca2+1002173021/144060d2ba+863872/2205db5-5430665495/24696db3a+175\ 1209/49dba2-4b7-772/49b5a-2016b3a2+2447488/15435c2+2892256/2205cdb-874774\ 1/70cb3+17152/245cba+33193268/15435db4+49664/2401db2a+208/35b6-736/49b4a-\ 576b2a2+3946912/3087cd-8711357/245cb2+9807232/1715ca+100/441d2b+9239464/2\ 205db3-27136871347/216090dba+3088/49b5-432/1715b3a-1152ba2-3501522/49cb+4\ 4256944/36015db2+2944/49b4-1984/245b2a+16325536/7203db+8645328/1715b3+889\ 6/1715ba+7936/245b2+4903616/1715b ==> J[14]=b9+5198192/5145c2d+12991037712583744/7004233215c2b2-105275530427811\ 674305/706026708072cd2b-865832/5145cdb3+6495749504301632/1000604745cdba+3\ 84/35cb5+3072/1715cb3a+1682791046032/50421cba2+10552/1701d2b4-21057561463\ 286141225/706026708072d2b2a-4/441db6-1504/108045db4a-1536/245db2a2+4b8-96\ /245b6a-16607680/352947b2a3+25981782536421632/7004233215c2b+1146484600/19\ 4481cd2+7936/36015cdb2+23968768/252105cda-167877614888/15882615cb4+16384/\ 2401cb2a-9216/7ca2-21104/11907d2b3+8013122344/6806835d2ba-664/15435db5-80\ 2341460/21609db3a+2074976/343dba2+36/7b7-512/343b5a-1757184/12005c2+256/1\ 08045cdb-5185904/245cb3+33655822382912/252105cba+1051727741306/200120949d\ 2b2+16596304/108045db4+1299114755104960/1400846643db2a+1504/245b6-512/343\ b4a-1024/21609cd-10353376/1715cb2-63238144/12005ca-640/3087d2b+33201824/1\ 08045db3-5358169136/252105dba+3728/343b5-6656/12005b3a-4149952/343cb+9789\ 44576/15882615db2+841395575432/7203b4-8704/12005b2a+8854528/50421db-55276\ 736/12005b3+2048/12005ba+16827911191456/252105b2-31619072/12005b ==> 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 ```