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arXiv:0806.0479v1[math.AC]3Jun2008Gr¨obnerbasesforthepolynomialringwithinfinitevariablesandtheirapplicationsKei-ichiroIimaandYujiYoshinoAbstractWedevelopthetheoryofGr¨obnerbasesforidealsinapolynomialringwithcountablyinfinitevariablesoverafield.Asanapplicationwereconstructsomeoftheone-onecorrespondencesamongvarioussetsofpartitionsbyusingdivisionalgorithm.1IntroductionThepurposeofthispaperistodevelopthetheoryofGr¨obnerbasesforidealsinapolynomialringk[x1,x2,...]withcountablyinfinitevariablesoverafieldk.Insuchacase,idealsarenotnecessarilyfinitelygenerated,andhencetheGr¨obnerbasesforidealsmightbeconsistingofinfinitepolynomials.HoweverweshallclaimthatthereisstillanalgorithmtogettheGr¨obnerbaseforagivenideal.ThisideaofGr¨obnerbasesforinfinitelygeneratedidealsisstronglymotivatedbythefollowingobservation.Recallthatasequenceλ=(λ1,λ2,...,λr)ofpositiveintegersiscalledapartitionofanon-negativeinte-gerniftheequalityλ1+λ2+···+λr=nholdsandλ1≥λ2≥···≥λr≥1.Insuchacasewedenoteitbyλ⊢n.Weareconcernedwiththefollowingsetsofpartitions:A(n)={λ⊢n|λi≡±1(mod6)},B(n)={λ⊢n|λi≡±1(mod3),λ1λ2···λr},C(n)={λ⊢n|eachλiisodd,andanynumberappearsinλi’satmosttwotimes}.ItisknownbythefamousSchur’sequalities(see[1])thatallthesesetsA(n),B(n)andC(n)havethesamecardinalityforalln∈N.Itisalsoknownthat1theone-to-onecorrespondencesamongthesethreesetsarerealizedinsomecombinatorialwayusing2-adicor3-adicexpansionsofintegers.Howeversuchone-to-onecorrespondencescanbereconstructedthroughthedivisionalgorithmbyusingthetheoryofGr¨obnerbases.Forthis,weneedtoex-tendthetheoryofGr¨obnerbasestoapolynomialringwithinfinitelymanyvariables.InSection1,weshallgivenecessarydefinitionsofinitialideals,Gr¨obnerbases,S-polynomialsandregularsequencesinthepolynomialringk[x1,x2,...].AndwedevelopthetheoryofGr¨obnerbasesforidealsinsuchapolynomialringbypresentingasequenceofpropositions,mostofwhichgoesinparallelwiththeordinarycaseforidealsinpolynomialringswithfinitelymanyvariables.ButthedifferenceisthatidealsarenotnecessarilyfinitelygeneratedandweneedtoargueaboutinfinitesetofpolynomialsasGr¨obnerbasesandregularsequences.OneoftheessentiallynewresultsinthispaperisTheorem1.12wherewegiveanalgorithmtogetthereducedGr¨obnerbases.TheotheroneisTheorem1.22inwhichweshowthatanypermutationofahomogeneousregularsequenceofinfinitelengthisagainaregularsequence.InSection2,weapplythetheorydevelopedinSection1tothesetsofpartitions.ThemainresultisTheorem2.1,wherewegiveone-to-onecor-respondencesbetweenvarioussetsofpartitionsbyusingdivisionalgorithminthetheoryofGr¨obnerbases.AsoneoftheapplicationsweshallgivethebijectivemappingamongtheabovementionedsetsA(n),B(n)andC(n).1.1Gr¨obnerbasesforidealsThroughoutthispaper,letkbeanyfieldandletS=k[x1,x2,...]beapolynomialringwithcountablyinfinitevariables.WedenotebyZ(∞)≥0thesetofallsequencesa=(a1,a2,...)ofintegerswhereai=0forallibutfinitenumberofintegers.AlsowedenotebyMon(S)thesetofallmonomialsinS.Sinceanymonomialisdescribeduniquelyasxa=Qixaiiforsomea=(a1,a2,...)∈Z(∞)≥0,wecanidentifythesesets,i.e.Mon(S)∼=Z(∞)≥0.IfweattachdegreeonSbydegxi=di,thenamonomialxahasdegreedegxa=P∞i=1aidi.Intherestofthepaper,weassumethatthedegreesdi’sarechoseninsuchawaythatthereareonlyafinitenumberofmonomialsofdegreedforeachd∈N.Forexample,thesimplestwayofattachingdegreeisthatdegxi=iforalli∈N.Definition1.1.AtotalorderonMon(S)iscalledamonomialor-derif(Mon(S),)isawell-orderedset,anditiscompatiblewiththe2multiplicationofmonomials,i.e.xaxbimpliesxcxaxcxbforallxa,xb,xc∈Mon(S).(See[4,Chapter15].)Notethattheorderx1x2x3···isnotacceptableformonomialorder,sinceitviolatesthewell-orderingcondition.Ontheotherhand,ifwearegivenanymonomialorder,then,renumberingthevariables,wemayassumethatx1x2x3···.ThefollowingareexamplesofmonomialordersonMon(S).(See[4,Chapter15,pp.329–330].)Example1.2.Leta=(a1,a2,...)andb=(b1,b2,...)beelementsinZ(∞)≥0.(1)Thepurelexicographicorderplisdefinedinsuchawaythatxaplxbifandonlyifaibiforthelastindexiwithai6=bi.(2)Thehomogeneous(resp.anti-)lexicographicorderhl(resp.hal)isdefinedinsuchawaythatxahlxb(resp.xahalxb)ifandonlyifeitherdegxadegxbordegxa=degxbandaibiforthelast(resp.first)indexiwithai6=bi.(3)Thehomogeneous(resp.anti-)reverselexicographicorderhrl(resp.harl)isdefinedasfollows:xahrlxb(resp.xaharlxb)ifandonlyifeitherdegxadegxbordegxa=degxbandaibiforthefirst(resp.last)indexiwithai6=bi.Asintheordersin(2)to(3),ifitsatisfiesthatdegxadegxbimpliesxaxb,thenwesaythattheorderishomogeneous.ThemonomialordersinExample1.2arealldistinctasshowninthefollowingexampleinwhichdegxi=ifori∈N:x4hlx1x3hlx22hlx21x2hlx41,x41halx21x2halx1x3halx22halx4,x4hrlx22hrlx1x3hrlx21x2hrlx41,x41harlx21x2harlx22harlx1x3harlx4.NowsupposethatamonomialorderonMon(S)isgivenandwefixitintherestofthissection.Then,anynon-zeropolynomialf∈Sisexpressedasf=c1xa(1)+c2xa(2)+···+crxa(r),whereci6=0∈kandxa(1)xa(2)...xa(r).Insuchacase,theleadingterm,theleadingmonomialandtheleadingcoefficientoffaregivenrespectivelyasℓt(f)=c1xa(1),ℓm(f)=xa(1)andℓc(f)=c1.Foranon-zero3idealI⊂S,theinitialidealin(I)ofIisdefinedtobetheidealgeneratedbya