TROPICALSCHEMETHEORY9TropicalschemesidealsoftropicalLaurentpolynom

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1 TROPICALSCHEMETHEORY9.Tropicalschemes,id
TROPICALSCHEMETHEORY9.Tropicalschemes,idealsoftropical(Laurent)polynomialsemirings,andvaluatedmatroidsWewilldiscusstherelationbetweentheTropicalschemes,idealsoftropical(Lau-rent)polynomialsemirings,andvaluatedmatroids.Theorem9.1.Letkbeaeldandletv:k!Tbeavaluation.ForY(k)nasubschemedenedbyanidealIk[x11;:::;x1n],anyofthefollowingobjectsdeterminetheothers.(1)ThecongruenceB(tropI)=B(v(I))onT[x11;:::;x1n],(2)TheidealtropI=ftropfjf2Ig,and(3)Theset(tower)ofvaluatedmatroidsofthevectorspaceIhd,whereIhk[x0;:::;xn]isthehomogenizationofIandIhdisthedegreedpartofIh.Homogenization :Thehomogenizationofapolynomialf=Pcuxu2k[x1;:::;xn]is~f=Xcuxu
x�juj+deg(f)0;wherejuj=u1+


+unanddegf=maxfjujjcu6=0g.Forexample,thehomogenizationoff=x21x32+x3is~f=x21x32+x3x40.ThehomogenizationofatropicalpolynomialF=Pauxu=min(au+x
u)is~F=min(au+x
u+(deg(F)�juj)x0):ThehomogenizationofanidealIk[x11;:::;x1n]isIh=f~fjf2I\k[x11;:::;x1n]g;whichisanidealofk[x0;x1;:::;xn].ThehomogenizationofarelationFGwithF;G2T[x1;:::;xn](i.e.supp(F);supp(G)Nn)withdegFdegGis~F~G+(deg(F)�deg(G))x0:ThehomogenizationofacongruenceJT[x ]T[x ]isthecongruenceJh=D^FG(F;G)2JE:Proposition9.2.LetIk[x11;:::;x1n]beanideal.ThenB(trop(Ih))=(B(tropI))h. Date:October17,2017,Speaker:KalinaMincheva,Scribe:NetanelFriedenberg.1 2TROPICALSCHEMETHEORYBeforeweproceedwiththeproofofTheorem9.1werecallthedenitionofatropicallinearspace:Denition9.3.LetMbeamatroidonanitesetE=f0;1;:::;ng.Thetropicallinearspacetrop(M)isthesetofvectors(w0;:::;wn)2Rn+1suchthatforanycircuitCofMtheminimumofthewisisattainedatleasttwiceasirangesoverC.ProofTheorem9.1,part1.Her

2 eweshowthat(1)and(2)determineeachother.(
eweshowthat(1)and(2)determineeachother.(Thesecondpartoftheproof,showingthat(2)and(3)determineeachother,ispostponeduntilthediscussionofvaluatedmatroidsandtropicallinearspaces,below.)ItiseasytoseethattropIdeterminesB(tropI).Conversely,byProposition9.2wehavethatB(tropI)determinesB(tropIh).Wecanregardanyhomogeneouspolynomialf2IdasalinearformonA(n+dd)whosecoordinatesareindexedbymonomialsofdegreedink[x1;:::;xn].Similarly,wecanregardatropicalpolynomialFofdegreedasatropicallinearformonT(n+dd).IfLdisthesubspaceofA(n+dd)wherethelinearformslfvanishforallf2Id,thentrop(Ld)=ld:=nz2T(n+dd)troplf(z)attainsitsminimumatleasttwiceo=nz2T(n+dd)troplf(z)=troplfbu(z)(8f2Ihd)(8u2supp(f))o=nz2T(n+dd)ltropf(z)=ltropfbu(z)(8f2Ihd)(8u2supp(f))o=nz2T(n+dd)lf(z)=lg(z)�8(f;g)2B(tropIhd)
o:ItfollowsthatB(tropIh)determinesthetowerfldgd0.Butldalsodeterminesitsduallinearspacel?d=trop(L?d).NotethatavectorliesinL?difandonlyifitisacoecientvectorforapolynomialinIhd.Thusthetropicallinearspacel?disthesameastrop(Ihd)(thedegreedpartoftrop(Ih)).ThisimpliesthatB(tropIh)determinestropIh.SincewecanrecoverIasIhjx0=1,thisdeterminestropI.Weproceedtointroducevaluatedmatroidsandexplaintheirconnectiontotropi-callinearspaces.Valuatedmatroidsandtropicallinearspaces LetEanitesetandr2N.Let�Er
denotethecollectionofsubsetsofEofsizer.Avaluatedmatroid MonEofrankrisafunctionp:�Er
!R[f1g,calledthebasisvaluationfunction,suchthat(1)9B2�Er
suchthatp(B)6=1and(2)foranyB;B02�Er
,forallu2BnB0thereexistsv2B0nBsuchthatp(B)�p(B0)p((Bnfug)[fvg)+p((B0nfvg)[fug).Thesupportofp,denedtobesupp(p)=fB2�Er
jp(B)6=1g,isthecollectionofbasesofarankrmatroidonE.Wecallthismatroid\theunderlyingmatroidofM"anddenoteit

3 byM . TROPICALSCHEMETHEORY3DenotebyMdthe
byM . TROPICALSCHEMETHEORY3DenotebyMdthesetofallmonomialsofdegreed.Iffisahomogeneouspoly-nomialofdegreedink[x0;:::;xn],thenwecanthinkoffasalinearformlfonthevectorspaceVdwithbasisMd.LetIdbethedegreedpartofahomogeneousidealIk[x0;:::;xn].ThenconsiderLd=fy2Vdjlf(y)=0(8f2Id)gVd.Wehaveapairingh�;�i:k[x0;:::;xn]dVd!kgivenbyhf;yi=lf(y).ThenthespaceLdistheannihilatorofId.LetdimLd=:rd.AnotherwaytoviewLdisasapointontheGrassmannianGr(rd;Vd)PN,whereN=�jMdjrd
�1.ThevaluatedmatroidM(Id)ofIdisthefunctionpd:�Mdrd
!R[f1gdenedbylettingpd(B)bethevaluationofthePluckercoordinateofLdindexedbyB.The(matroid-theoretic)vectorsofM(Id)aretropicalpolynomials.VectorsofminimalsupportarecircuitsofM(Id).ForexampleF=min(a1+xu1;a2+xu2),fromwhichwegetthevectorfu1;u2goftheunderlyingmatroid.ProofofTheorem9.1,part2.Wecannowshowthat(2)and(3),i.e.,trop(I)andfM(Id)gd0determineeachother.Asdiscussedabovetheelementsoftrop(I)darethevectorsofthevaluatedmatroidM(Id),sotrop(I)determinesandisdeterminedbythesetofvaluatedmatroidsfM(Id)gd0.Theorem9.4.LetY(k)nbethesubschemedenedbyanidealIk[x11;:::;x1n].Thenthemultiplicitiesofthemaximalcellsoftrop(Y)canberecoveredfromB(tropI).ThiscanbethoughtofasthetropicalHilbert-Chowmorphism.Multiplicitiesintropicalgeometry Werecallthedenitionofmultiplicitiesforthemaximalcellsofatropicalvariety.LetYbeasubvarietyof(k)nofdimensiond.Letv:k!�beavaluationwhichadmitsasplitting�!k,i.e.agrouphomomorphismw7!twsuchthatv(tw)=w.Firstrecallthattrop(Y)isapolyhedralcomplexofpuredimensiond.Forw2Rnandf=Pcuxutheinitialformoffwithrespecttowisinwf=Xval(cu)+w
u=trop(f)(w)t�val(cu)cuxu2k[x1;:::;xn];wherekistheresidueeld.WecansimilarlydenetheinitialidealinwI=hin

4 wfjf2Ii.Themultiplicityofwisdenedto
wfjf2Ii.Themultiplicityofwisdenedtobemult(w)=XPaminimalassociatedprimeofinwImult(P;inwI);wheremult(P;inwI)isthemultiplicityofPinaprimarydecompositionofinwI.Afterasuitablemonomialchangeofcoordinateson(k)n,wecanshowthatinwIisgeneratedbypolynomialsinxd+1;:::;xn.Thenmult(w)=dimkk[x1d+1;:::;x1n] inwI\k[x1d+1;:::;x1n].Indeed,oneobservesthatthemultiplicitiesdonotchangewhenwepasstok[xd+1;:::;xn]andthentheproblemisreducedtocomputingprimarydecompositionofthezeroideal 4TROPICALSCHEMETHEORYintheArtiniank-algebrak[x1d+1;:::;x1n] inwI\k[x1d+1;:::;x1n](cf.Lemma3.4.7in[MS]).Todenethemultiplicityofamaximalcell,pickwintherelativeinteriorof.Thenmult()=mult(w).Inordertorecoverthemultiplicitiesfromthebendrelations,weneedtoextendGrobnertheorytocongruences.LetF=Puauxu=min(au+x
u)2T[x0;:::;xn].Forw2Rn+1thewedenetheinitialformofFwithrespecttowtobeinwF:=minau+w
u=F(w)x
u2B[x0;:::;xn].ForG=min(bu+x
u)2T[x0;:::;xn]weconsidertherelationFG.TheinitialformofFGisdenedtobeinw(FG):=((inwFinwG)ifF(w)=G(w)(0inwG)ifF(w)�G(w).Denition9.5.IfCisacongruencetheninwC=hinw(FG)jFG2Ci.Example9.6.LetF=min(0+x;1+y;2+z).F^x=min(1+y;2+z).SoFF^xisinB(F).Forw=(2;1;3),wehaveinwF=min(x;y)andinwF^x=y.Soinw(FF^x)=min(x;y)y.Proposition9.7(Tropicalizationandinitialformscommute).(a)Forf2k[x11;:::;x1n]andw2Rn,inw(tropf)=trop(inwf).(b)ForIk[x11;:::;x1n]andw2Rn,inw(B(tropI))=B(trop(inwI)).Werstobservetheeectofachangeofcoordinatesontropicalvarietiesandcongruences.Amonomialchangeofcoordinateson(k)n,x u 7!x Au forsomeA2GLn(Z)correspondstoamapF(x )7!F(ATx )=:A
F,whereFisatropicalpolynom

5 ial(cf.Lemma3.2.7in[MS]).ForacongruenceC
ial(cf.Lemma3.2.7in[MS]).ForacongruenceConT[x11;:::;xn1]wedeneA
C=fA
FA
Gj(FG)2Cg.Onecanshowthatthisactioncommuteswithtropicalization:trop(V(A
I))=A
trop(V(I)).ProofofTheorem9.4.LetY(k)nwithdeningidealIandletdimY=dimtrop(Y)=d:Wepickwintherelativeinteriorof,whereisamaximalcelloftrop(V(I)).LetL=span(w�w0jw02).AfterachangeofcoordinateswecanassumethatL=span(e1;:::;ed).ByLemma3.3.6in[MS]itfollowsthatwehavethatL=tropV(inwI)=V(trop(inwI)):Indeed,if:=trop(V(I))=fwjinwI6=(1)gandamaximalcellandwintherelativeinteriorofwehaveL=star(w)=fvjinv(inw(I))6=(1)g.WeknowfromProposition9.7thatLcanberecoveredfromB(trop(inwI))=inw(B(tropI)).SoLcanberecoveredfromB(tropI).TheinitialidealinwIishomogeneouswithrespecttothegradingdeg(xi)=eifor1idanddeg(xi)=0otherwise.Asremarkedearlier,thereisageneratingsetink[x1d+1;:::;x1n]forinwIk[x11;:::;x1n]. TROPICALSCHEMETHEORY5ConsiderinwI\k[x1;:::;xn]anddenotebyinwIhthehomogenizationofinwIink[x0;:::;xn].Recallthatmult()=dimkk[x0;xd+1;:::;xn] inwIh\k[x0;xd+1;:::;xn].How-ever,k[x0;xd+1;:::;xn] inwIh\k[x0;xd+1;:::;xn]iszero-dimensionalsotheHilbertpolynomialofk[x0;xd+1;:::;xn] inwIhmustbeaconstantpolynomial.RecallthattheHilbertpolynomialofahomogeneousidealJcanberecoveredfromB(tropJ).Toshowthatmult(w)canberecoveredfromB(tropI)itisenoughtoshowthatB(trop(inwIh))canberecoveredfromB(tropI).FromProposition9.2andProposition9.7weknowthatB(trop(inwIh))=B(trop(inwI))handB(trop(inwI))h=inw(B(tropI))h:TheseimplythatB(trop(inwIh))=inw(B(tropI))hwhichcanberecoveredfromB(tropI).References[MS]D.Maclagan,B.SturmfelsIntroductiontotropicalgeometry,Volume161ofGraduateStud-iesinMathematicsAMS