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1、THETROPICALj-INVARIANTERICKATZ,HANNAHMARKWIG,THOMASMARKWIGAbstract.If(Q,A)isamarkedpolygonwithoneinteriorpoint,thenageneralpolynomialf∈K[x,y]withsupportAdefinesanellipticcurveCfonthetoricsurfaceXA.IfKhasanon-archimedeanvaluationintoRwecantropicalizeCfto
2、getatropicalcurveTrop(Cf).IftheNewtonsubdivisioninducedbyfisatriangulation,thenTrop(Cf)willbeagraphofgenusoneandweshowthatthelatticelengthofthecycleofthatgraphisthenegativeofthevaluationofthej-invariantofCf.1.IntroductionPreviousworkbyGrishaMikhalkin[1
3、3],byMichaelKerberandHannahMarkwig[11]andbyMagnusVigeland[18]showsthatthelengthofthecycleofatropicalcurveofgenusonehaspropertieswhichoneclassicallyattributestothej-invariantofanellipticcurvewithoutgivingadirectlinkbetweenthesetwonumbers.In[9]weestablis
4、hedsuchadirectlinkforplanecubicsbyshowingthatthetropicalizationofthej-invariantisingeneralthenegativeofthecyclelength.Inthepresentpaperwegeneralizethisresulttoellipticcurvesonothertoricsurfacesusingthesamemethods.Moreprecisely,if(Q,A)isamarkedpolygonwi
5、thoneinteriorpoint,thenageneralpolynomialf∈K[x,y]withsupportAdefinesanellipticcurveCfonthetoricsurfaceXA.IfKhasanon-archimedeanvaluationwecantropicalizeCftogetatropicalcurveTrop(Cf).IftheNewtonsubdivisioninducedbyfisatriangulation,thenTrop(Cf)willbeagra
6、phofgenusoneandweshowinourmainresultinTheorem6.4thatthelatticelengthofthecycleofthegraphisthenegativeofthevaluationofthej-invariant.arXiv:0803.4021v2[math.CO]31Mar2008Inthecasewherethetriangulationisunimodular,i.e.allthetriangleshavearea1,2thisresultwa
7、sindependentlyderivedbyDavidSpeyer[17,Proposition9.2]usingTateuniformizationofellipticcurves.DavidSpeyer’sresultismoregeneralthoughinthesensethatitappliestocurvesinarbitrarytoricvarieties.Thispaperisorganizedasfollows.InSection2weconsidertoricsurfacesd
8、efinedbyamarkedlatticepolygonwithoneinteriorpoint,werecalltheclassificationofthesepolygonsandweconsidertheimpactonthej-invariantforthecorrespondingellipticcurves.Section3recallsthenotionoftropicalizationandofplanetropicalcurves.Wethenintr
