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1、Thegeneraltheoryoflineardi erenceequationsovertheinvertiblemax-plusalgebraNaliniJoshi(nalini@maths.usyd.edu.au)andChrisOrmerod(chriso@maths.usyd.edu.au)SchoolofMathematicsandStatisticsF07,TheUniversityofSydneyAbstract.Wepresentthemathematicaltheoryunderlyingsystems
2、oflineardi er-enceequationsovertheinvertiblemax-plusalgebra.TheresultprovidesananalogueofisomonodromytheoryforultradiscretePainleveequations,whichareextendedcel-lularautomata,andprovideevidencefortheirintegrability.OurtheoryisanalogoustothatdevelopedbyBirkho andhi
3、sschoolforq-di erencelinearequationsbutstandsindependentlyofthelatter.AsanexamplewederivelinearproblemsinthisalgebraforultradiscreteversionsofthesymmetricPIVequationandshowhowitactsastheisomonodromicdeformationofthelinearsystem.Keywords:Integrablesystems,cellularau
4、tomata,monodromy,ultradiscrete,Painleve,TropicalMathematicalSubjectClassi cation(2000):39A20,14H70,16Y601.IntroductionThediscretePainleveequationsareintegrableinthesensethattheycanbesolvedthroughassociatedsystemsoflinearequations.Fordi erenceequationsthelinearthe
5、orynecessaryforsolvabilitywasdevelopedbyBirkho [1]andhisschoolandimprovedbyRamisandhisschool[2].Thepurposeofourpaperistoprovidesuchatheoryforlinearultradiscreteequations,whichcanberegardedasequationsposedovertheinvertiblemax-plusalgebra.TheclassicalresultsofBirkho
6、wereextendedforsystemsofq-di erenceequationsin[3,4,5].RamisandhisschoolimprovedtheseresultsformanycasesanddevelopedanalyticGaloistheoryforq-di erenceequations[2].ThediscretePainleveequationshavebeenanareaofintenserecentresearch[6].JimboandSakai[7]werethe rsttoappl
7、yBirkho stheorytoaq-di erenceversionofthesixthPainleveequation.Inthispaper,weconsiderultradiscreteversionsofsuchdiscretePainleveequations.Ultradiscreteequationsareobtainedthroughalimitingprocessin-troducedin[8].TheultradiscretePainleveequationscanbeinterpretedas
8、integrablecellularautomata[6,9,10].UltradiscreteanaloguesofintegrableequationshavebeenshowntohaveLaxPairsoverthemax-plussemiring[11]and[12].Wecon
