Algebraic Condition for Integrable Numerical Algorithms

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1 Algebrac Codto or Itegrable Numercal Algorthms TOSHIAKI ITOH Mathematcal ad Natural Sceces Itegrated Arts ad Sceces The Uversty o Tokushma Tokushma Tokushma JAPAN Abstract: - Algebrac Numercal Algorthm ANA) especally Algebrac Fte Derece Equato AFDCE) are treated by algebrac geometrcally. By coheret shea ad proper morphsm codto [] we ca treat tegrablty o ANA ad AFDCE. I ths work we treated tegrable propertes o Lear AFDCE ad Durad -Kerer -Aberth method rom ths codto as examples. Key-Words: - Algebrac te derece equato Coheret shea Aalytc space Itegrablty Itroducto Crtera or tegrablty o dscrete dyamcal system especally or the tegrablty o o-lear te derece equato are proposed recetly [2-3]. Usg these crtera we ca d out ew o-lear AFDCEs that are caddates o tegrable AFDCEs IAFDCEs). These crtera are called Sgularty Coemet ad Algebrac Etropy crtera respectvely. Though we ca get IAFDCEs by these crtera we have ot yet had clear theoretcal backgroud or these crtera. O the other had algebrac treatmet o dscrete evolutoal equatos cotrol system was attempted [4]. I ths approach cocepts o coheret sheaves were troduced to treat dscotuous dyamcal systems properly. From these works GAGA prcple was troduced to AFDCE [ ]. Though the prcple s abstractve or practcal applcato we have to traslate t to proper statemet usg orthodox umercal otato. The ext ew sectos are prepared or ths purpose. I these sectos we wll troduce ope covergs by Zarsk topology ad sheaves ad coheret sheaves by deals rom AFDCE. By these preparatos we ca dee IAFDCEs as algebrac evolutoal equatos o whch soluto uctos are holomorphc uctos aalytc space by several complex varables except or approprate sgulartes. I ths work we troduce Ste space [] as aalytc space ad proper space as algebrac space or AFDCE. GAGA prcple coects aalytc space ad algebrac space. From ths pot AFDCEs are uctoal equatos aalytc space. Smple ad mportat algebrac space whch correspods to aalytc Ste) space s proectve space. Thereore we ca corm some valdty o sgularty coemet crtero. Moreover we ca uderstad backgroud o algebrac etropy crtero rom ths approach. 2 Coheret sheaves by AFDCE ae space Algebrac traslatos o AFDCE are show ths secto. Cosder ollowg smple 2-step algebrac te derece equato as a example F + here s teger ad F - + ) C[ - + ]. C[ - + ] s polyomal ucto by { - + } wth complex coecet C. We wrte C C[ - + ] ad F F +. Here ) C ad meas order o the sequece o pots { }. The we ca regard as uctoal equato o ). It s kow C s Noethera cossts o te umber o deals). Cosder localato by treatg F as deal C. I F s rreducble polyomal prme deal) C the C \F becomes multplcatve set C. We put S C \F ad A. The A ad F A become S C local rg ad maxmal deal. A F A ) s localato ad A /F A s ucto eld. We dee X SpecA ad X as all prme deal o A. We call X as ae scheme at. We ca cosder morphsm ϕ : X X. Ths s also a kd o map or coecto betwee X ad X + that should satsy some codto or tegrablty.

2 Dscrete aalogy or sheaves o modules AFDCE to usual ae scheme ca be obtaed as ) Assume every F correspods prme deals. I F s t prme deal we decompose t to prme deals rst. ) Make A rom F. The A s Noethera locally at least because Ideals o A gve sub-algebra o C[ - + ]. Clearly C[ - + ] s Noethera by Hlbert s bass theorem thereore each A s Noethera. A becomes O F modules. Notato O F meas local quotet rg ad ucto eld. ) Treat each φ : A A s homomorphsm by atural morphsm { - + } { + +2 }. I ths codto s broke we must mody F.. v) Dee X A){Collecto o all X A )}.We troduce Zarsk topology to X A) by ope coverg U ad D that are deed as U {p p p X} ad D {p F p p X}. Here XSpecA{Collecto o all SpecA } X SpecA. We d X s Noethera locally because A s Noethera. Usg above detos we ca troduce sheaves o AFDCEs. It s kow that sheaves by Ideals become coheret sheaves. Proper scheme over C whch s coheret shea correspods to some aalytcal scheme by GAGA. Especally proectve scheme over C s proper scheme. Thereore valdty o sgularty coemet crtero s oud. It s atural codto rather tha crtero proectve space. Usg ths prcple more actually we show what mplemetato o AFDCE satses property o shea especally coheret shea ad becomes proper scheme by ollowg some examples. Sce Z collecto o all A ad C C[ - + ] C[ + +2 ] C[ k- k k+ ] are polyomals cosst o te umber o varables. Thereore Hlbert s bass theorem s ot satsed globally. That s SpecA has te elemets ad ot Noethera. Please remember that prevous mplemetato ) to v) satsy coheret shea codto oly locally at every ). Thereore we eed more codtos to costruct etre coheret shea o AFDCE by ths ormulato. For the codto o te umber o varables etre space we must add more codto or AFDCE. For example orcg ollowg codto gves Noethera property o etre space o AFDCE F F 2) F + 2 F or all F F or equvaletly F F or all. Sce by mplct ucto theorem we ca d ollowg local relatos at every 3) + g +) - ) - g -) + ) here g +) g --) should be algebrac ucto ad ever spol algebrac property o each F. We kow that ths codto ca be moded to the case o g +) ad g --) are holomorphc but we leave t. The we ca delete - ad +2 rom C[ - + ] C[ + +2 ] as C[g -) + ) + ] C[ + g ++) + )]. Applg ths codto or all F we d all C[ - + ] C[ + +2 ] C[ k- k k+ ] are cluded the two varable polyomal C[ k k+ ] or holomorphc ucto. Sce k s arbtrary we ca say C[ k k+ ] s germ at k k+ ad also represetato o soluto ucto o AFDCE by germ at k k+. It s easy to geerale ths treatmet or mult-step several varable ad smultaeous AFDCE. Wth ths codto ths example XSpecA O X collecto o A )) becomes coheret shea etrely. Expresso o C[ + ] by C[ k k+ ] k s aalogous to Taylor seres represetato C[ + ] by { k k+ }. I ths case t correspods uctoal seres represetato or ear eghbor uctos. We also oud the codto 2) correspods to preservg dmeso o varables each A. +2 A + takes over depedecy o - A or tal codtos are preserved rom A to A +. Deto: We call AFDCE that satses coheret shea codtos as coheret AFDCE CAFDCE). We call these codtos as coheret codto or abbrevato. For geeral mult-step several varables or smultaeous ADFCE we dee coheret codto as ) F gves coordate rg ad F geerates covergs o AFDCE as a o-sgular algebrac maold. Moreover A becomes Noethera at every. ) Exstece o proper morphsm A A + at every ad every A satses coheret codto by Zarsk topology [8]. ) Followg dmesoal codto s satsed depedetly o each coverg wth regular coordate system. dma )dmital codtos or Boudary codtos) Cost. Deto: We call sgular pot set) o AFDCE where coheret codto s broke. It s clear rom the deto that CAFDCE has o sgular pot set). I other words t becomes o-sgular algebrac maold usg proper local coordates.

3 2. IAFDCE example Lear AFDCE Cosder -sep lear AFDCE ad ts soluto 4) Cs y Cs y + Cs y + L + Cs y y + c y + c2 y + + c y 5) c L here Cs c ad y k are Casorata tegral costat ad udametal solutos. Usually tegral costats are oted as C -c /c ad Casorata Cs s deed as matrx determat by elmatg colum rom ollowg + matrx 2 3 y y y L y 2 3 y y y L y 6) 2 3 y y y y L 2 M M M L M 2 3 y y y L y Let F Cs y + Cs y + Cs2 y2 + L+ Cs y. It s clear A s Noethera because the umber o elemets y... ) are te. Moreover F satses codto or te umber o total varables because F / y Cs ad F / y Cs. Codtos Cs ad Cs due to udametal solutos y... ). By ths property we ca rewrte 4) as y Cs y + Cs2 y2 + L + Cs y ) / Cs or y Cs y Cs y Cs y ) + + L + / Cs. We oud smlarty betwee 4) ad 5) comes drectly rom algebra o coheret property. Algebrac sgularty o F s gve by F / y ad F / y. Clearly F s o-sgular because Cs ad Cs. Lear AFDCE s a typcal model equato that relects coheret shea structure well. Note that Lear AFDCE satses te algebrac property. We ca easly corm ths property rom 7) yr { h Cs... Cs ys + h Cs... Cs ys+ + L + h Cs... Cs ys+ }/ h Cs... Cs. I ths case arbtral y r ca be represeted by a geeral algebrac relato 7) wth arbtral germ at {s s+ s+-} ad the umber o germs s depedet o posto. 3 Proectve scheme AFDCE We troduced coheret codto to AFDCE. The shea codto clude cocept o coecto betwee each coverg. Tradtoal ways o aalyss or AFDCE also pay atteto or the coecto; although t hasgored the algebrac te property utl ow. We ever overlook vestgato o te property or AFDCE ater ths because algebrac te property s the ma cocept o coheret shea. I addto we use GAGA to gve tegrablty to AFDCE we ever overlook proper morphsm property o AFDCE. It s kow that morphsm proectve space s proper morphsm thereore we do t eed to pay atteto to ths property whe we treat AFDCE proectve space. From ths act we oud proectve space s smple ad coveet space or applyg GAGA to AFDCE. I ths secto we revew proectve scheme shortly or ths purpose. We ca d more detals rom may textbooks o algebrac geometry. We assume all AFDCEs ths secto are homogeeous equatos. As a example usg the same otato prevous secto we treat F C. I ths case C C [ ; ; + ; o ] correspods to polyomal ucto wth complex coecet proectve space. The F s deed the subspace o exteded proectve space by ollowg treatmet 8) + { + } { ; ; ;} becto here ;;;} { ; ; ; } ad { + 9) F ) m F + ) m whe total order o F - + ) equals m. By ths treatmet we ca regard F proectve space as ) B ) m F + ) G - ; ; + ; ) here C ; ; ; ] s homogeeous B [ + o equato ad F G - ; ; + ;. Thereore we ca regard B as homogeeous deal. For smplcty we assume B s a homogeeous prme deal. We cosder a space ProPA ) whch cossts o all homogeeous prme deals except or rrelevat deal quotet rg PA S C here S C [ ; ; +; o ] \ B. We call ths space PX ProPA ). As the same maer ae space we ca troduce Zarsk topology locally usg ollowg detos or ope coverg D {p PA p p PX}.

4 We also use ae coverg U to cover D. I ths case set o U s er coverg tha set o D. I U U Φ we treat 2) ) m F ) ) m F m m ) ) m F + ) ) ) ). ) We ca treat cluso F to proectve space by deret way rom prevous example as ollowg. I U U U Φ + 3) + { + } { ; ; ;;; } becto here +... { ;;;;;} { ; ; + ; ; ; + 4) F } ad - ) m ) m2 + ) m3 F + ) + G - ; ; + ; - ; ; + ) m m2 m3 order o - + F. Coecto betwee each coverg at U U U ) U U U Φ ) ca be deed by the same way as prevous example. We use otato PX{Collecto o all ProPA )}. Note that PX ProPA ) becomes tely geerated O B -module because B s deed by F ad A ad A s clearly tely geerated O F -module. A becomes te coverg o B. Moreover graded rg PA s Noethera. The ProPA ) becomes coheret shea at locally. We must add more codto to PX whch becomes coheret shea globally addto to 2). At the rst we must dee a rule how to choose proper or all. Clearly we have o rule yet or selectg or all. We must choose the total umber o s te. Istead ProPA ) becomes ot tely geerated space. It maybe also proper choce or to make PA o-sgular algebrac maold or example s deed rom blowg-up at each. We must also assume s tely geerated. For the purpose we assume aother relato or example 5) F o - + ). I addto ) or 4) also satses codto same to 2) ad 3). More complex case s also cosderable or example 6) F o ). ) Shortly ad 5) or 6) should orm local coecto or each ad. Whe these codtos are satsed by ew AFDCE system ad each PX cossts o te umber o geerator by homogeeous elemet the the ew AFDCE system gves codto or ProPA) whch becomes coheret shea. I ths case each PA becomes o-sgular algebrac maold wth te umber o ae coverg A thereore we d PX by { ; ; + ; o } or { ; ; + ; ; ; + } becomes regular local rg. Regular rg gves approprate local parameters or the algebrac maold; at last they spa regular coordate rg. I ths example dmeso o each local base space at + s eght by { } wth our relatos {G G + F F + }. We expect dmpropa )) dmpropa + )) 4 because { + + } should become te umber o base elemet or germ at +. It s clear dmpropa )4 also correspods to umber o tegral costats or tal codtos at +. I other words arbtrary ad ca be regarded as ucto C[ + + ] PA or holomorphc ucto by { + + }. We d dvergece o some varables } AFDCE whch ca be properly { + treated by space ; ; ; } by usg local { + o ae coverg because t s proper morphsm by coheret codto. The dvergece o AFDCE s oud oly part o ae coverg space. Note that resoluto or blowg-up procedure s ecessary to make above coverg. At preset we have o automatc blowg-up ad dow algorthm. Thereore algebrac etropy crtero becomes a kd o prescrpto or ths problem at preset. 4 Covergece ad tegrablty o ANA We treat orthodox umercal algorthm as a sample applcato usg prevous results. Durad -Kerer -Aberth method s umercal root dg algorthm or algebrac equato. Cosder -th degree algebrac equato wth real umber coecet 7) P ) + a + L + a a here C. The umber o roots ca be obtaed umercally by ollowg Newto s method

5 k ) L ) ) k ) ) M M M k ) L ) ) J ) ) / ) k + k ) k) k) 8) J ) ) k s terato umber. The 8) ca be wrtte as 9) k + k ) k ) k ) k ) P ) / ) We ca easly d that 9) s holomorphc mappg k) ) except or the case k. Usually we ca k ) ) assume k at every terato step thereore we ca regard 9) as holomorphc mappg at aytme. Clearly 7) has umbers o costats whch are equals to gve a. Especally a s varat or k that s 2) k) r) a k r.. From the same algebrac treatmet to AFDCEs 9) gves umbers o geerators or deals. It s k ) + clear that each equato or k ) 9) s depedet thereore they become geerator o deals. We ca troduce ope coverg by Zarsk topology as show Fg. 22) ϕ k) : k ) k+ k ) ) s proper mappg [8 2] wheever k s satsed. From these acts we ca say that Durad-Kerer-Aberth method satses coheret ad proper codtos. It s CAFDCE algorthm ad geerates tegrable system step-by-step by sel-tegrable deormato. Moreover gvg approprate tal codto whch gratees covergece correspods to gvg some deormed tegrable system. Ths property may gve superor coverget property o Durad-Kerer-Aberth method. Note that ths deormato s ot reversble as to k because the deormato s cotractve by covergece property. 5 Cocluso Algebrac treatmets o AFDCEs ad ANAs are show. It became clear that sgularty coemet ad algebrac etropy crtera are some parts o codtos o coheret ad proper morphsm codtos related to GAGA. Moreover sample AFDCEs whch satsy coheret codto ad gve proper morphsm are gve. By these samples smple but actual treatmet o AFDCEs ad applyg possbltes to aalye orthodox ANAs usg proposed codto are show. Fg. Algebrac vew o Durad-Kerer-Aberth method We ca dee restrcto mappg o k+ k ) k + k ) D D D D φ as Z k ) k+m k) k ) + 2 ρ : D D k k + k + D k+ k k k- - I ths case t s clear that mappg k ) D k D Reereces: [] T. Itoh Coheet Shea Codto or Fte Derece Equatos Comm. Math. Phys. to be publshed. [2] V. Papageorgou F. W. Nho B. Grammatcos ad A. Rama Isomoodoromc Deormato Problems or Dscrete Aalogues o Palevé Equatos Phy. Lett. A Vol pp [3] Falqu G. ad Vallet c. M. Sgularty Complexty ad Quas-Itegrablty o Ratoal mappg Comm. Math. Phys Vol pp [4] V. Lomade M. S. Rav J. Rosethal ad J. M. Schumacher A behavoral approach to sgular systems Report MAS-R988 Sept [5] R. C. Gug ad H. Ross Aalytc Fuctos o Several Complex Varables Pretce-Hall Ic [6] T. Itoh Dscretato or Ordary Deretal Equatos that have Exact Solutos It. Jour. Appl. Math Vol. No pp

6 [7] J.P. Serre Géométre algébrques et géométre aalytque A. Ist. Fourer Vo pp [8] R. Hartshore Algebrac Geometry Sprger- Verlag 977. [9] K. Ueo ad M. Dasuu-kka 23 Algebrac Geometry) Iwaam-Shote 999 Japaese. [] P. Grths ad J. Harrs Prcples o Algebrac Geometry Wley-Iterscece Pub [] H. Grauert ad K. Frtsche Several Complex Varables Sprger-Verlag 976. [2] W. Read Udergraduate Commutatve Algebra Cambrdge Uv. Press 995. Appedx GAGA prcple Theorem Serre): Let X be a proper proectve) scheme over C. The the uctor h duces a equvalece o categores rom the category o coheret sheaves o X to the category o coheret aalytc sheaves o X h. Furthermore or every coheret shea I o X the atural maps α : H X I) H X h Ih ) are somorphsms or all.