Metaptuqiako Programmasth Logikh kai Jewria Algorijmwn kai Upologismouµ∏

λ∀Trigwnopoi sei kai ApalefousaDiplwmatikh ergasiaQr sto Konax Epiblèpwn: Iwnnh Z. Emrh , An. Kajhght ,Tm ma Plhroforik kai Thlepikoinwni¸n, EKPA

Aj na, IoÔlio 2006

Sth mhtèra mou.

Given free will but within certain limitations,

I cannot will myself to limitless mutations,

I cannot know what I would be if I were not me,I can only guess me. .

Robert Wyatt

PerieqìmenaEisagwg 71 Jewra Algebrik Apaloif 91.1 Klasik (probolik ) Apalofousa . . . . . . . . . . . . . . . 91.2 Torik (toric) Apalofousa . . . . . . . . . . . . . . . . . . . 111.2.1 PolÔtopa, ajrosmata Minkowski kai meiktì ìgko . . 121.2.2 Orismì kai idiìthte th torik apalofousa . . . . 152 Trigwnopoi sei 212.1 Eisagwg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2 Kukl¸mata . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.3 DeutereÔon PolÔtopo . . . . . . . . . . . . . . . . . . . . . . 263 Aparjmhsh twn kanonik¸n trigwnopoi sewn 283.1 Tèqnasma Cayley . . . . . . . . . . . . . . . . . . . . . . . . . 283.2 O algìrijmo aparjmhsh . . . . . . . . . . . . . . . . . . . . 303.2.1 O algìrijmo th antstrofh anaz thsh . . . . . . . 313.2.2 Perigraf tou algorjmou aparjmhsh twn kanonik¸ntrigwnopoi sewn . . . . . . . . . . . . . . . . . . . . . 334 Aparjmhsh twn diamorf¸sewn meikt¸n keli¸n 374.1 Eisagwg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.2 H eidik perptwsh th genik jèsh . . . . . . . . . . . . . . 384.3 H genik perptwsh . . . . . . . . . . . . . . . . . . . . . . . . 415 Mia efarmog sthn algebrikopohsh 475.1 Eisagwg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.2 O algìrijmo IPSOS . . . . . . . . . . . . . . . . . . . . . . . 475.3 Algìrijmo probol . . . . . . . . . . . . . . . . . . . . . . . 48Eplogo 50Bibliografa 513

Katlogo Sqhmtwn1.1 PolÔtopa tou NeÔtwna. . . . . . . . . . . . . . . . . . . . . . 121.2 'Ajroisma Minkowski enì trig¸nou kai enì tetrag¸nou. . . . 131.3 Kataskeu kanonik meikt upodiaresh . . . . . . . . . . . . 171.4 Meiktè kai mh meiktè upodiairèsei . . . . . . . . . . . . . . . 182.1 Antistrofè akm¸n pou odhgoÔn sthn i-tupik trigwnopohsh. 222.2 Paradegmata kuklwmtwn se distash 1, 2 kai 3. . . . . . . . 232.3 'Ena kÔklwma kai oi trigwnopoi sei tou. . . . . . . . . . . . . 242.4 Antistrofè akm¸n. . . . . . . . . . . . . . . . . . . . . . . . . 252.5 DeutereÔonta polÔtopa enì pentag¸nou kai enì tetrapleÔrou. 273.1 H idèa tou teqnsmato Cayley gia n = 1, 2. . . . . . . . . . . 293.2 Efarmog tou teqnsmato Cayley. . . . . . . . . . . . . . . . 303.3 Pardeigma efarmog antstrofh anaz thsh . . . . . . . . 324.1 H perptwsh dÔo poluwnÔmwn se mia metablht ktw apì thnupìjesh genik jèsh . . . . . . . . . . . . . . . . . . . . . . . 404.2 H morf ekfulismènwn kuklwmtwn gia d = 2, 3 kai k = 1, 2. . 414.3 To kÔklwma tou paradegmato 4.3.1. . . . . . . . . . . . . . . 424.4 H genik perptwsh dÔo poluwnÔmwn se mia metablht . . . . . 46

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PerlhyhTo antikemeno th ergasa aut enai h aparjmhsh twn klsewn isoduna-ma meikt¸n keli¸n mia oikogèneia sunìlwn shmewn kai o upologismì toupolutìpou tou NeÔtwna th apalefousa enì sust mato poluwnÔmwn.Uprqei mia 1-1 kai ep antistoiqa twn klsewn aut¸n, me ti korufè tou polutìpou tou NeÔtwna th apalefousa tou sust mato poluwnÔmwnme sÔnola st rixh sthn oikogèneia aut . H antistoiqa aut ma epitrèpei naupologsoume ta akraa mon¸numa (extreme monomials) th apalefousa .Mèsw tou teqnsmato Cayley, h aparjmhsh ìlwn twn klsewn isoduna-ma meikt¸n keli¸n, angetai sthn aparjmhsh ìlwn twn klsewn isodunama twn kanonik¸n trigwnopoi sewn enì nèou sunìlou. H aparjmhsh twn te-leutawn mpore na gnei me efarmog diadoqik¸n topik¸n metasqhmatism¸n(bistellar flips) se katllhla uposÔnola xekin¸nta apì mia arqik trigwno-pohsh.Protenetai h tropopohsh enì apotelesmatikoÔ algorjmou antstrofh anaz thsh [14, ¸ste autì na upologzei mìno ti klsei isodunama twnkanonik¸n trigwnopoi sewn.Tèlo , dnetai mia efarmog twn parapnw mejìdwn me thn perigraf enì algorjmou algebrikopohsh [7, o opoo upologzei èna upersÔnolo twn mo-nwnÔmwn th algebrik exswsh mia parametrik dosmènh (uper)epifneia .

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Euqariste Euqarist¸ ton epiblèponta kajhght mou, k. Iwnnh Emrh, giat me èfere seepaf me èna sunarpastikì kai sqedìn gnwsto s' emèna pedo, gia ti upo-dexei kai thn suneq st rix tou. Euqarist¸ epsh ta mèlh th trimeloÔ epitrop , kajhghtè k. Hla Koutsoupi kai k. Euggelo Rpth. Stonteleutao ofelw èna idiatero euqarist¸ gia th suneq enjrrunsh kai thnempistosÔnh tou. Qwr autè den ja tan dunat h sunèqish twn spoud¸nmou.Euqarist¸ ìlou tou suntelestè tou MetaptuqiakoÔ Progrmmato sthLogik kai thn jewra Algorjmwn kai idiatera tou kajhghtè k. G. Mo-sqobkh kai k. K. Dhmhtrakìpoulo gia th sumbol tou sth dhmiourga enì filikoÔ kai episthmonik karpofìrou klmato .Tèlo euqarist¸ ton patèra mou kai tou flou mou gia thn upomon kaithn st rix tou kaj' ìlh th dirkeia twn spoud¸n mou.Aj na, Qr sto Konax IoÔlio 2006

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Eisagwg To kÔrio antikemeno melèth sthn ergasa aut enai to polÔtopo tou NeÔtw-na th torik arai apalofousa enì uperprosdiorismènou sust mato poluwnÔmwn me sumbolikoÔ suntelestè kai h anptuxh algorjmwn gia tonupologismì tou. Perigrfontai basik stoiqea th jewra arai apaloif kaj¸ kai ta ergalea apì th sunduastik gewmetra pou aut qrhsimopoie.H jewra arai apaloif xeknhse apì thn ergasa twn Gelfand, Kapra-nov kai Zelevinsky [9 to 1994, ìpou eis gagan thn melèth diakrinous¸n kaiapaloifous¸n me sunduastik ergalea. Apì tìte to pedo èqei melethje kaianaptuqje èntona kai efarmìzetai sthn eplush poluwnumik¸n susthmtwn,sthn apaloif metablht¸n, sthn algebrikopohsh kai alloÔ. 'Ena pardeigmaefarmog sthn algebrikopohsh dnoume sto keflaio 5 me thn perigraf toualgorjmou IPSOS [7, tropopoihmènou ¸ste na qrhsimopoie ta apotelèsmatath ergasa aut . Sugkekrimèna o algìrijmo qrhsimopoie mia ulopohshtou J. De Loera [5, h opoa upologzei ìle ti kanonikè trigwnopoi sei enì sunìlou shmewn. AntikajistoÔme aut thn ulopohsh me mia nèa poubaszetai sthn antstrofh anaz thsh kai upologzei klsei isodunama ka-nonik¸n trigwnopoi sewn. Autì èqei w apotèlesma thn drastik mewsh th poluplokìthta q¸rou tou algorjmou algebrikopohsh .H klasik probolik apalofousa enì sust mato n poluwnÔmwn f1,

. . . , fn se n metablhtè me sumbolikoÔ suntelestè , enai èna polu¸numome akèraiou suntelestè kai metablhtè tou suntelestè twn poluwnÔmwn,to opoo ma parèqei mia sunj kh gia thn epilusimìthta tou sust mato twnfi. Sugkekrimèna h probolik apalofousa mhdenzetai an kai mìno an tapolu¸numa pou antistoiqoÔn se mia anjesh tim¸n stou sumbolikoÔ sunte-lestè , èqoun koin rza. Me lla lìgia gia sust mata sugkekrimènh dom h opoa kajorzetai apì tou bajmoÔ twn poluwnÔmwn, h probolik apalo-fousa apotele èna krit rio gia ti timè twn suntelest¸n pou antistoiqoÔnse sust mata ta opoa èqoun lÔsh. Gia ton upologismì th probolik apa-lofousa kataskeuzontai pnake h orzousa twn opown isoÔtai me thnapalofousa me èna pollaplsiì th . Oi pnake auto en dunmei mporoÔnna perièqoun opoiod pote mon¸numo bajmoÔ mikrìterou sou tou bajmoÔkje poluwnÔmou kai autì auxnei shmantik to mègejì tou . Epiplèon horzousa twn pinkwn aut¸n mpore na mhdenzetai tautotik me apotèlesma namhn orzetai h probolik apalofousa. Se kpoie apì autè ti peript¸sei 7

enai dunatìn na oriste h torik apalofousa.H torik arai apalofousa exetzei thn Ôparxh riz¸n ston q¸ro (C∗)n.GenikeÔei ta apotelèsmata th klasik jewra apaloif me thn ènnoia ìtitautzetai me thn probolik apalofousa gia sust mata pukn¸n poluwnÔmwn.Se antjesh me ton qarakthrismì twn poluwnÔmwn mèsw tou bajmoÔ tou ,h jewra arai apaloif ekmetalleÔetai thn dom tou jewr¸nta ta w ajrosmata monwnÔmwn. H dom aut antanakltai gewmetrik apì to po-lÔtopo tou NeÔtwna. An antistoiq soume kje mon¸numo me mh mhdenikìsuntelest enì poluwnÔmou f , me to dinusma twn ekjet¸n tou, tìte topolÔtopo tou NeÔtwna N(f) tou poluwnÔmou enai to kurtì perblhma twndianusmtwn aut¸n. To polÔtopo autì parèqei pollè plhrofore sqetikme thn gewmetrik dom th uperepifneia f = 0. Sugkekrimèna h asumptw-tik sumperifor th uperepifneia aut kajorzetai apì ti korufè toupolutìpou N(f). Epiplèon to polÔtopo tou NeÔtwna ekfrzei thn araiìth-ta tou poluwnÔmou. H ekmetlleush th dom twn poluwnÔmwn odhge semikrìterou pnake th apalofousa kai epomènw se beltwsh th polu-plokìthta upologismoÔ th .Mia apì ti pio entupwsiakè efarmogè twn polutìpwn tou NeÔtwna sealgebrik probl mata ofeletai stou Bernstein, Kouchnirenko kai Khovan-sky oi opooi èdwsan frgmata ston pl jo twn lÔsewn poluwnumik¸n su-sthmtwn qrhsimopoi¸nta meiktoÔ ìgkou . To apotèlesma autì genikeÔeièna klasikì apotèlesma tou Bezout to opoo parèqei frgmata sto pl jo twn lÔsewn pou ekfrzontai apì to ginìmeno twn bajm¸n twn poluwnÔmwn.O bajmì th torik apalofousa sqetzetai epsh me to frgma tou Bern-stein.EpalhjeÔonta thn sten sqèsh th jewra arai apaloif me th sun-duastik gewmetra, èna shmantikì apotèlesma [9, 18 enai h perigraf toupolutìpou tou NeÔtwna N(R), th torik apalofousa R, mèsw twn trigw-nopoi sewn enì sunìlou shmewn. Sugkekrimèna mporoÔme na upologsoumeti korufè tou polutìpou N(R) upologzonta merikè mìno apì ti tri-gwnopoi sei tou polutìpou tou NeÔtwna enì bohjhtikoÔ poluwnÔmou. TopolÔtopo tou NeÔtwna autoÔ tou bohjhtikoÔ poluwnÔmou mpore na kata-skeuaste kai mèsw th apeikìnish Cayley ìpw perigrfetai sthn enìthta3.1. To apotèlesma autì ekmetalleuìmaste sto keflaio 4, tropopoi¸nta ènan algìrijmo aparjmhsh kanonik¸n trigwnopoi sewn [12, ¸ste na apa-rijme mìno klsei isodunama twn kanonik¸n trigwnopoi sewn mèsw twnopown mporoÔme na anakt soume ti korufè tou N(R).

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Keflaio 1Jewra Algebrik Apaloif Sto keflaio autì perigrfontai sunoptik ta basik stoiqea th jewra algebrik apaloif (elimination theory).1.1 Klasik (probolik ) ApalofousaH apalofousa Sylvester resultant R = Res(f, g, x), enì sust mato duopoluwnÔmwn f, g ∈ C[x], mef = a0x

n + . . . + an, a0 6= 0, n > 0,

g = b0xm + . . . + bm, b0 6= 0, m > 0,orzetai w h orzousa tou (n + m) × (n + m) pnaka Sylvester:

S =

a0 b0

a1 a0 b1 b0

a2 a1. . . b2 b1

. . .... a2. . . a0

... b2. . . b0

an

... . . . a1 bm

... . . . b1

an a2 bm b2. . . ... . . . ...an bm

O pnaka Sylvester emfanzetai sth bibliografa kai w o anstrofo touparapnw pnaka.Ja lème ìti to G enai èna polu¸numo stou suntelestè twn f0, f1, . . . , fnan gia kje suntelest cij twn fi uprqei mia metablht uij sto G. MeG(f0, f1, . . . , fn) sumbolzoume ton arijmì pou parnoume antikajist¸nta k-je metablht uij me ton antstoiqo suntelest cij tou fi.9

H apalofousa R = Res(f, g, x) enai èna polu¸numo me akèraiou sunte-lestè , me metablhtè tou (sumbolikoÔ ) suntelestè twn f, g , pou mhden-zetai ìpote ta f, g èqoun koin rza. To ìnom th ofeletai sto gegonì ìti hRes(f, g, x) apalefei th metablht x apì to sÔsthma twn f, g. Ta parapnwsumfwnoÔn me tou orismoÔ 1.1.2 kai 1.2.6, pou dnoume sth sunèqeia.H apalefousa R = Res(f, g, x) mpore na ekfraste me polloÔ trì-pou . Gia pardeigma an x1, . . . , xm enai oi rze tou poluwnÔmou g, tìte hapalefousa dnetai apì ton tÔpo

R = Res(f, g, x) = am0

m∏

i=1

g(xi). (1.1)MporoÔme na epektenoume thn ènnoia th apalofousa kai gia polu¸numame perissìtere apì mia metablhtè .Orismì 1.1.1. O probolikì q¸ro Pn enai to sÔnolo twn klsewn iso-dunama twn dianusmtwn sto C

n + 1 me toulqiston mia mh mhdenik sunte-tagmènh, ìpou tautzoume ta pollaplsia me èna mh mhdenikì stajerì arijmìl ∈ C

∗. Dhlad :P

n :=(c1, c2, . . . , cn+1)∈Cn+1|∃i ci 6= 0&(c1, c2, . . . , cn+1) ≡ l(c1, c2, . . . , cn+1), l∈C

∗.'Estw F0, F1, . . . , Fn, n + 1 omogen polu¸numa me sumbolikoÔ sunte-lestè uij , sunolikoÔ bajmoÔ d0, . . . , dn antstoiqa, ston C[x], ìpou x =x0, . . . , xn. To sÔsthma twn n + 1 exis¸sewn se n + 1 agn¸stou

F0(x) = F1(x) = · · · = Fn(x) = 0 (1.2)èqei pnta th mhdenik lÔsh (x0, . . . , xn) = (0, . . . , 0). Ma endiafèrei nagnwrzoume pìte èqei kai lle lÔsei ektì th mhdenik . Thn apnthshsto er¸thma autì ma thn dnei h apalofousa tou sust mato (1.2):Orismì 1.1.2. H probolik apalofousa R = Res(F0, F1, . . . , Fn) enaièna polu¸numo me akèraiou suntelestè , me metablhtè tou (sumbolikoÔ )suntelestè uij twn F0, F1, . . . , Fn , gia to opoo isqÔei R = 0 an kai mìnoan to sÔsthma (1.2) èqei mh mhdenik koin lÔsh sto Pn.Je¸rhma 1.1.1. [9 H apalofousa R = Res(F0, F1, . . . , Fn) tou sust -mato (1.2) enai to monadikì angwgo polu¸numo ston Z[uij ], gia to opooisqÔei R = 0 ann to sÔsthma (1.2) èqei mh mhdenik koin lÔsh sto P

n.Pardeigma 1.1.1. 'Estw to sÔsthma grammik¸n exis¸sewnF0 = c00x0 + · · · + c0nxn = 0... (1.3)Fn = cn0x0 + · · · + cnnxn = 0.10

To sÔsthma (1.3) èqei monadik lÔsh an kai mìno an h orzousa tou pnakatwn suntelest¸n tou enai sh me mhdèn. H orzousa aut enai èna polu¸numostou suntelestè twn Fi kai tautzetai me thn apalofousa tou sust mato .O tÔpo 1.1 mpore na genikeute kai sthn perptwsh omogen¸n poluwnÔ-mwn F0, F1, . . . , Fn ∈ C[x0, x1, . . . , xn], bajm¸n d0, . . . , dn antstoiqa, me thnupìjesh ìti Res(F 0, F 1, . . . , Fn−1) 6= 0:R = Res(F0, F1, . . . , Fn) = Res(F 0, F 1, . . . , Fn−1)

dn∏

p∈V

fn(p)m(p), (1.4)ìpoufi(x0, . . . , xn−1) = Fi(x0, . . . , xn−1, 1),

F i(x0, . . . , xn−1) = Fi(x0, . . . , xn−1, 0),

V = V (f0, . . . , fn−1), to sÔnolo lÔsewn twn poluwnÔmwn f0, . . . , fn−1 (dh-lad to omoparallhlikì (affine) tm ma tou probolikoÔ algebrikoÔ sunìlouV (F0, . . . , Fn−1)) kai m(p) h pollaplìthta th rza p ∈ V .H apalofousa enì sust mato omogen¸n poluwnÔmwn upologzetai etew to phlko th orzousa enì katllhlou tetragwnikoÔ pnaka pro thnorzousa enì upopinak tou [13, ete w o mègisto koinì diairèth enì su-nìlou orizous¸n katllhlwn pinkwn. Stìqo enai h kataskeu tetragwni-k¸n pinkwn elqisth distash , opìte h apalofousa enai h orizous tou .Genik den uprqei tÔpo gia ton upologismì th apalofousa se sunrthshtwn suntelest¸n tou sust mato . Se merikè peript¸sei enai dunatì oupologismì th w orzousa enì mìno pnaka kai tìte lème ìti o tÔpo enaideterminental. Sti upìloipe peript¸sei epidi¸ketai h kataskeu pinkwnme to mikrìtero dunatì mègejo .1.2 Torik (toric) ApalofousaUprqoun sust mata twn opown h klasik apalofousa enai tautotik mh-denik . Autì sumbanei ìtan uprqei èna sÔnolo riz¸n sto peiro tou opoouh distash enai jetik . Se autè ti peript¸sei mia diaforetik apalefou-sa mpore na oriste, h opoa kai ja parèqei èna krit rio gia thn Ôparxh mhlÔsewn tou sust mato .H klasik apalofousa pou orsame sthn prohgoÔmenh enìthta, aforse pukn (dense) polu¸numa, ta opoa perièqoun ìla ta mon¸numa sunolikoÔbajmoÔ sou me to sunolikì bajmì tou poluwnÔmou 1 kai parèqei plhrofore 1Akribèstera, èna polu¸numo se n metablhtè enai puknì ìtan èqei polÔtopo touNeÔtwna pou enai Minkowski pollaplsio tou monadiaou aplìkou ston R

n, bl. OrismoÔ 1.2.2 kai 1.2.3. 11

gia thn Ôparxh probolik¸n riz¸n. Sth prxh ìmw , suqn sunantme po-lu¸numa pollo apì tou suntelestè twn opown enai mhdeniko. Tìte enaishmantikì na oriste mia torik (toric) arai (sparse) apalofousa, h opo-a ekmetalleÔetai th dom twn sugkekrimènwn poluwnÔmwn. En¸ h klasik apalofousa ekfrzei thn Ôparxh probolik¸n riz¸n (dhlad riz¸n sto Pn),h torik apalofousa ekfrzei thn Ôparxh torik¸n riz¸n, dhlad riz¸n semia torik pollaplìthta (manifold) T th opoa o tìro (C∗)n enai puknìuposÔnolo, (C∗)n ⊆ T ⊆ P

m, ìpou m enai to pl jo twn monwnÔmwn ìlwntwn poluwnÔmwn.1.2.1 PolÔtopa, ajrosmata Minkowski kai meiktì ìg-ko Orismì 1.2.1. To sÔnolo st rixh (support) A ⊂ Zn enì poluwnÔmou

f ∈ C[x1, . . . , xn], enai to sÔnolo twn dianusmtwn twn ekjet¸n twn mo-nwnÔmwn me mh mhdenikì suntelest tou f . An A = a1, a2, . . . , ak, tìtef =

∑ki=1 cix

ai , ìpou ci 6= 0,∀i = 1, . . . , k.'Ena polÔtopo enai to kurtì perblhma enì peperasmènou sunìlou sh-mewn.Orismì 1.2.2. 'Estw èna polu¸numo f ∈ C[x1, . . . , xn] me sÔnolo st -rixh A ⊂ Zn. To polÔtopo tou NeÔtwna N(f) tou poluwnÔmou f enai tokurtì perblhma conv(A) tou sunìlou st rix tou.To polÔtopo tou NeÔtwna enì poluwnÔmou antikatoptrzei gewmetrikthn dom tou kai apotele to ergaleo gia thn melèth algebrik¸n idiot twnmèsw th sunduastik gewmetra .Pardeigma 1.2.1. 'Estw to puknì deuterobjmio polu¸numo g(x, y) =

c0 + c1x + c2y + c3x2 + c4xy + c5y

2 kai to araiì polu¸numo epsh deutèroubajmoÔ, f(x, y) = a1x − a2y − a3xy. Sto Sq ma 1.1 blèpoume ta polÔtopatou NeÔtwna twn poluwnÔmwn f, g. Enai fanerì pw to polu¸numo f èqeipolÔ mikrìtero polÔtopo tou NeÔtwna apì to polu¸numo g.N(f)

N(g)

Sq ma 1.1: Ta polÔtopa tou NeÔtwna gia to pardeigma 1.2.1.Akolouj¸nta mia antstrofh diadikasa, dojènto enì sunìlou A =a1, a2, . . . , ak ⊂ Z

n, mporoÔme na orsoume to sÔnolo twn poluwnÔmwn,twn opown ta mon¸numa èqoun ekjète sto A:12

L(A) = c1xa1 + · · · + ckx

ak | ci ∈ C.To sÔnolo L(A) enai èna dianusmatikì q¸ro ep tou C distash k.Efìson to sÔnolo A mpore na perièqei kai dianÔsmata me arnhtikè sun-tetagmène , prèpei na epektenoume tou orismoÔ twn monwnÔmwn kai poluw-nÔmwn ¸ste na èqoume kai ta antstoiqa algebrik antikemena. Autì odhgesthn ènnoia twn Laurent monwnÔmwn kai poluwnÔmwn:'Estw a = (a1, . . . , an) ∈ Zn, èna dinusma me akèraie suntetagmène . To an-tstoiqo Laurent mon¸numo sti metablhtè x1, . . . , xn enai to xa = xa1

1 . . . xann .Ta Laurent polu¸numa enai peperasmènoi grammiko sunduasmo Laurent mo-nwnÔmwn. To sÔnolo twn Laurent poluwnÔmwn enai èna metajetikì daktÔ-lio me ti sun jei prxei th prìsjesh kai tou pollaplasiasmoÔ poluw-nÔmwn kai sumbolzetai C[x±1

1 , . . . , x±1n ].Gia th sunèqeia ja qreiastoÔme merikè gewmetrikè ènnoie .Orismì 1.2.3. 'Estw A, B ⊂ R

n kai λ ≥ 0 pragmatikì arijmì .a. To jroisma Minkowksi A + B twn duo sunìlwn enaiA + B = a + b | a ∈ A & b ∈ B ⊂ R

n.b. To bajmwtì ginìmeno tou A me to λ enaiλA = λa | a ∈ A ⊂ R

n.An ta A, B enai polÔtopa, tìte kai to A + B enai polÔtopo kai epiplèonmpore na upologiste w h kurt j kh tou sunìlou a+b | a ∈ V ert(A), b ∈V ert(B), ìpou me V ert(P ) sumbolzoume to sÔnolo twn koruf¸n tou polu-tìpou P .

+ =A1A2

A +A1 2

Sq ma 1.2: 'Ajroisma Minkowski enì trig¸nou kai enì tetrag¸nou.MporoÔme na epektenoume tou parapnw orismoÔ kai gia opoiod pote(peperasmèno) pl jo sunìlwn polutìpwn.To jroisma Minkowski twn polutìpwn tou NeÔtwna duo poluwnÔmwn f, gantistoiqe sto polÔtopo tou NeÔtwna tou ginomènou f · g, dhlad N(f) +N(g) = N(f · g).Mia apì ti pr¸te efarmogè twn polutìpwn tou NeÔtwna tan sthneÔresh tou arijmoÔ twn koin¸n riz¸n enì sust mato poluwnumik¸n exis¸-sewn. 13

'Estw Vol(P) o eukledio ìgko tou polutìpou P ⊂ Rn orismèno ètsi¸ste o monadiao kÔbo ston R

n na èqei monadiao ìgko. Gia tuqaa polÔtopaP1, . . . , Pn kai λ1, . . . , λn ∈ R+, h èkfrash Vol(λ1P1 + · · · + λnPn) enai ènaomogenè polu¸numo sti metablhtè λ1, . . . , λn, bajmoÔ n, [10.Orismì 1.2.4. O meiktì ìgko MV (P1, . . . , Pn) twn polutìpwn P1, . . . , Pn ⊂R

n enai o suntelest tou grammikoÔ ìrou, dhlad tou ìrou λ1 . . . λn, toupoluwnÔmou V ol(λ1P1 + · · · + λnPn), jewroÔmeno w polu¸numo sti meta-blhtè λ1, . . . , λn.'Ena isodÔnamo orismì tou meiktoÔ ìgkou twn polutìpwn P1, . . . , Pnenai o epìmeno .Orismì 1.2.5. 'Estw P1, . . . , Pn ⊂ Rn polÔtopa kai λ1, . . . , λn ∈ R+.Uprqei mia monadik pragmatik sunrthsh MV (P1, . . . , Pn) gia thn opoaisqÔoun ta ex :a. H MV (P1, . . . , Pn) enai grammik w pro kje metablht , dhlad

MV (P1, . . . , λkPk+µP′

k, . . . , Pn) = λMV (P1, . . . , Pk, . . . , Pn)+µMV (P1, . . . , P′

k, . . . , Pn),b. MV (P1, . . . , Pn) = n! · Vol(P ), an P1 = · · · = Pn = P .Pìrisma 1.2.1. MV (P1, . . . , Pn) ≥ 0. Isìthta isqÔei an to ∑Pi èqeimhdenikì ìgko (isodÔnama distash mikrìterh tou n) an uprqei kpoio Pitètoio pou dim(Pi) = 0.Pìrisma 1.2.2. H MV (P1, . . . , Pn) mpore na upologiste apì ton tÔpo(egkleismoÔ - apokleismoÔ):MV (P1, . . . , Pn) =

∑

I⊂1,...,n

(−1)n−|I|Vol

(∑

i∈I

Pi

). (1.5)Sth sunèqeia ja exetsoume frgmata ston arijmì twn koin¸n riz¸n ka-l orismènwn (tetrgwnwn) susthmtwn poluwnÔmwn. 'Estw èna sÔsthmapoluwnÔmwn

f1 = f2 = · · · = fn = 0, fi ∈ C[x±11 , . . . , x±1

n ], (1.6)kai Pi = N(fi), i = 1, . . . , n, ta polÔtopa tou NeÔtwna twn poluwnÔmwn fi.An ta sÔnola st rixh twn poluwnÔmwn fi tautzontai, to sÔsthma kaletaimh meiktì, diaforetik kaletai meiktì.To pr¸to frgma pou ja exetsoume dnetai apì to je¸rhma tou Bezoutkai afor sti koinè rze twn poluwnÔmwn ston Cn ston P

n.Je¸rhma 1.2.3. (Bezout)[4 An to sÔsthma (1.6) èqei peperasmèno arijmìlÔsewn kai kama lÔsh sto peiro, tìte èqei to polÔ d = d1 · d2 . . . dn lÔsei ston Cn, ìpou di o bajmì tou poluwnÔmou fi. Epiplèon an ta polu¸numa fienai genik (generic), tìte to frgma enai akribè .14

To epìmeno je¸rhma pou apodeqjhke apì ton D. Bernstein to 1975 geni-keÔei to je¸rhma tou Bezout. H perptwsh duo poluwnÔmwn se duo metablh-tè eqe dh apodeiqje apì ton Minding to 1841, prin o Minkowski eisgeithn ènnoia twn meikt¸n ìgkwn.Je¸rhma 1.2.4. (Bernstein BKK)[4, 9 An to sÔsthma (1.6) èqei pepe-rasmèno arijmì lÔsewn, tìte èqei to polÔ d = MV (P1, . . . , Pn) lÔsei ston(C∗)n. Epiplèon an ta polu¸numa fi enai genik (generic), tìte to frgmaenai akribè .1.2.2 Orismì kai idiìthte th torik apalofousa H arai torik apalefousa, orzetai gia uperprosdiorismèna sust mata th morf

f0 = f1 = · · · = fn = 0 ∈ C[x±11 , . . . , x±1

n ]. (1.7)Sth perptwsh aut den orzetai o meiktì ìgko tou sust mato . Mpo-roÔme ìmw na exgoume èna nw frgma ston arijmì twn koin¸n riz¸n tousust mato qrhsimopoi¸nta meiktoÔ ìgkou .Parat rhsh 1.2.1. An to sÔsthma (1.7) èqei peperasmèno arijmì lÔsewn,tìte èqei to polÔ d = MV−i(P0, . . . , Pi−1, Pi+1, . . . , Pn) lÔsei ston (C∗)n.Epiplèon an ta polu¸numa fi enai genik (generic), tìte to frgma enaiakribè .'Estw A0, A1, . . . , An ⊂ Zn tètoia pou to ∑n

i=0 Ai na pargei ton Zn kai

fi ∈ L(Ai), Laurent polu¸numa me sÔnola st rixh ta Ai. Kje fi enai th morf fi =∑

aij∈Aicijx

aij . Tautzoume kje polu¸numo fi me to dinusmatwn suntelest¸n tou, ci = (ci0, ci1, . . . , cin) kai to sÔsthma twn fi me todinusma c = (c0, c1, . . . , cn).JewroÔme to sÔnolo Z0 twn dianusmtwn c pou antistoiqoÔn se sust -mata poluwnÔmwn th morf (1.7) me fi ∈ L(Ai), kai ta opoa èqoun koin rza. 'Estw Z h Zariski kleistìthta tou sunìlou Z0. To sÔnolo Z enai ènaangwgo algebrikì sÔnolo. H torik apalefousa tou sust mato twn fienai to monadikì angwgo polu¸numo pou orzei thn Z.Orismì 1.2.6. H arai torik apalefousa R = Res(A0, A1, . . . , An)twn poluwnÔmwn f0, f1, . . . , fn enai to monadikì (mèqri pros mou) polu¸numome akèraiou suntelestè kai metablhtè tou sumbolikoÔ suntelestè twnfi, dhlad R ∈ Z[c], me ti akìlouje idiìthte :a) an codim(Z) = 1, tìte h R enai to monadikì angwgo polu¸numo pouorzei thn uperepifneia Z,b) an codim(Z) > 1, tìte R := 1.Pìrisma 1.2.5. To R enai omogenè polu¸numo w pro tou sumbolikoÔ suntelestè kje poluwnÔmou fi, bajmoÔ sou me MV−i(f0, . . . , fi−1, fi+1, . . . , fn).An ta polu¸numa fi èqoun koin rza ston (C∗)n, tìte R = 0.15

An ta polu¸numa fi enai pl rh, (isodÔnama ta sÔnola Pi = conv(Ai)enai Minkowski pollaplsia tou monadiaou aplìkou ston Zn), tìte h arai apalefousa tautzetai me thn klasik apalefousa.Orismì 1.2.7. H oikogèneia twn sunìlwn st rixh Ai twn poluwnÔmwn

fi, i = 0, . . . , n, ja kaletai ousi¸dh (essential) an gia to omoparallhlikìplègma (affine lattice)L :=

n∑

i=0

λiai | ai ∈ Ai, λi ∈ Z,

n∑

i=0

λi = 1

pou pargetai apì ta Ai isqÔei rank(L) = n kai rank(L′) ≥ |J | gia kjeJ ⊂ 0, 1, . . . , n, ìpou L′ enai to omoparallhlikì plègma pou pargetai apìthn oikogèneia sunìlwn st rixh Aj |j ∈ J. .An kje sÔnolo Ai èqei kurt j kh distash n, tìte h oikogèneia twnsunìlwn st rixh Ai enai ousi¸dh .Pìrisma 1.2.6. [19 'Estw h oikogèneia sunìlwn st rixh Aii∈I . Gia toalgebrikì sÔnolo Z (bl. Orismì 1.2.6) isqÔei codim(Z) = 1 an kai mìno anuprqei mia monadik upooikogèneia sunìlwn st rixh Aii∈J h opoa enaiousi¸dh . Tìte h torik apalofousa tou sust mato poluwnÔmwn me sÔnolast rixh ta Aii∈I tautzetai me thn torik apalofousa tou sust mato poluwnÔmwn me sÔnola st rixh Aii∈J .O upologismì twn torik¸n apaleifous¸n baszetai sthn kataskeu enì tetragwnikoÔ pnaka M tou opoou h orzousa ete isoÔtai me thn apalefousaete enai èna mh tetrimmèno pollaplsio th . To basikì ergaleo gia thkataskeu tou pnaka M enai oi meiktè upodiairèsei [Canny-Emiris.Orismì 1.2.8. Mia poluedrik upodiaresh enì polutìpou Q enai miasullog apì pl rou distash polÔtopa Qi ta opoa diamerzoun to Q kaian dÔo tèmnontai se mia koin tou ìyh (pijan¸ ken ).Orismì 1.2.9. 'Estw èna polÔtopo P = P1 + · · · + Pm ⊂ R

n dista-sh n. Mia meikt upodiaresh S tou P enai mia sullog apì polÔtopaRi, i = 1, . . . , k, distash n, ta keli mègisth distash th upodiaresh ,pou ikanopoioÔn ti paraktw idiìthte :1. Gia kje i 6= j, h Ri∩Rj enai mia koin èdra (pijan¸ ken ) twn Ri, Rj ,2. Ta Ri apoteloÔn mia diamèrish tou P , dhlad P = R1 ∪ · · · ∪ Rk,3. Kje kel Ri grfetai w Minkowski jroisma uposunìlwn twn Pj :

Ri = Fi1 + · · · + Fim, Fij ⊂ Pj kai isqÔei n = dim(P ) = dim(Ri) ≤dim(Fi1) + · · · + dim(Fim).An sthn prohgoÔmenh sqèsh isqÔei isìthta, tìte h meikt upodiaresh kaletailept (fine, tight). IsodÔnama, mia meikt upodiaresh kaletai lept an kjemeiktì kel th den perièqei w gn sio uposÔnolo tou èna llo meiktì kel.16

An S, S′ enai duo meiktè upodiairèsei enì polutìpou P = P1 + · · · +Pm ⊂ R

n, ja lème ìti h S ekleptanei (refines) thn S′ kai ja grfoume S ≤ S′,an gia kje meiktì kel R =∑

Fi th S, uprqei èna meiktì kel R′ =∑

F ′ith S′, ¸ste Fi ⊆ F ′

i gia kje i. To sÔnolo twn meikt¸n upodiairèsewn toupolutìpou R me th sqèsh eklèptunsh (≤), enai èna merik¸ diatetagmènosÔnolo (poset). Ta elaqistik stoiqea tou sunìlou autoÔ enai oi leptè meiktè upodiairèsei kai gi' autè isqÔei ìti kje kel tou enai Minkowskijroisma aplìkwn. To monadikì megistikì stoiqeo enai h tetrimmènh meikt upodiaresh P . Oi leptè meiktè upodiairèsei enai gia ti meiktè upodiai-rèsei to anlogo twn trigwnopoi sewn gia ti poluedrikè upodiairèsei .Mia meikt upodiaresh S tou polutìpou P = P1+ · · ·+Pn, enai kanonik (regular), an uprqei mia sunrthsh anÔywsh (lifting function) ω : P → R,tètoia pou h S na enai h probol twn ìyewn tou ktw (isodÔnama tou nw)peribl mato tou sunìlou ∑ Pi, ìpou Pi = (a, ω(a))|a ∈ Pi. To dinusmaw me suntetagmène ta ω(a), kaletai dinusma bar¸n (weight vector) th upodiaresh . An h sunrthsh anÔywsh ω enai genik (generic), tìte hepagìmenh kanonik meikt upodiaresh enai lept .Sth sunèqeia ja grfoume meikt upodiaresh enno¸nta kanonik lept meikt upodiaresh.

(a) (b)Sq ma 1.3: Kataskeu mia kanonik (a) kai mia ìqi kanonik (b) meikt upodiaresh mèsw mia sunrthsh anÔywsh .Orismèna keli mia meikt upodiaresh èqoun idiaterh shmasa.Orismì 1.2.10. 'Ena kel R = F1 + · · · + Fm, Fi ⊂ Pi, mia meikt upodiaresh tou polutìpou P = P1 + · · ·+ Pm ⊂ Rn kaletai meiktì kel an

dim(Fi) ≤ 1 gia kje i = 1, . . . , m.Epomènw , an m = n, dhlad P = P1 + · · · + Pn, tìte kje meiktìkel Ri grfetai w Minkowski jroisma akm¸n twn Pi (dim(Fi) = 1). Sthnperptwsh pou ta Pi enai ta polÔtopa tou NeÔtwna n+1 Laurent poluwnÔmwnf0, f1, . . . , fn se n metablhtè , kje meiktì kel ja enai th morf R =F0 + · · · + Fi + · · · + Fn, ìpou dim(Fi) = 0 kai dim(Fj) = 1, j 6= i, kaikaletai meiktì tÔpou i.To epìmeno je¸rhma deqnei th qrhsimìthta twn meikt¸n upodiairèsewn.17

Je¸rhma 1.2.7. [4 'Estw P1, . . . , Pn polÔtopa ston Rn kai S mia meikt upodiaresh tou ajrosmato Minkowski P = P1 + · · · + Pn. Tìte o meiktì ìgko MV (P1, . . . , Pn) upologzetai apì ton tÔpo

MV (P1, . . . , Pn) =∑

R

Vol(R), (1.8)ìpou to jroisma enai pnw se ìla ta meikt keli R th S.+ =A1

A2

Sq ma 1.4: Meiktè kai mh meiktè upodiairèsei .Mia llh efarmog twn meikt¸n upodiairèsewn tou ajrosmato Minkow-ski n + 1 polutìpwn, enai ston upologismì th torik apalofousa touantstoiqou sust mato poluwnÔmwn, ìpw anafèrjhke kai parapnw.Profan¸ , gia kje polÔtopo P = P0 + . . . + Pn uprqoun pollè mei-ktè upodiairèsei oi opoe kataskeuzontai apì diaforetikè sunart sei anÔywsh . Arketè apì autè èqoun ta dia meikt keli. Ja qwrsoume tosÔnolo twn meikt¸n upodiairèsewn tou P se klsei isodunama pou kaloÔn-tai diamorf¸sei meikt¸n keli¸n (mixed-cell configurations) [15, 18. Kjediamìrfwsh meikt¸n keli¸n perièqei ìle ti meiktè upodiairèsei oi opoe èqoun ta dia meikt keli, epomènw an endiaferìmaste mìno gia ta meiktkeli mia upodiaresh , arke na melet soume mìno ti diamorf¸sei meikt¸nkeli¸n.Uprqei mia sten sqèsh metaxÔ twn diamorf¸sewn meikt¸n keli¸n toupolutìpou P = P0 + · · · + Pn kai merik¸n monwnÔmwn. 'Estw h apalefousaR tou sust mato twn poluwnÔmwn

fi =∑

aj∈Ai

cijxaj , i = 0, . . . , n, j = 1, . . . , |Ai|, (1.9)ìpou N(fi) = Pi = conv(Ai).Ta mon¸numa th apalefousa ta opoa antistoiqoÔn se korufè toupolutìpou tou NeÔtwna N(R), kaloÔntai akraa (extreme monomials).H apalofousa R tou sust mato (1.9) enai èna polu¸numo stou sum-bolikoÔ suntelestè cij twn fi. Epomènw kje mon¸numo th R, enai th morf ∏ c

vi,a

i,a . MporoÔme na tautsoume to mon¸numo autì me to dinusmatwn ekjet¸n tou (. . . , vi,a, . . .). Gia kje sunrthsh sunrthsh anÔywsh 18

ω, upologzoume thn tim th sto dinusma twn ekjet¸n tou monwnÔmou. Htim aut enai to bro (weight) tou monwnÔmou w pro thn ω. H arqik morf (initial form), initω(R), th R enai to jroisma twn monwnÔmwn th me mègisto bro w pro thn ω [18.To epìmeno je¸rhma orzei mia apeikìnish tou sunìlou twn meikt¸n upo-diairèsewn ep tou sunìlou twn monwnÔmwn th arqik morf th apale-fousa . H apeikìnish aut gnetai 1-1 kai ep an thn periorsoume sto sÔnolotwn diamorf¸sewn meikt¸n keli¸n.Je¸rhma 1.2.8. [18 'Estw ìti ta sÔnola st rixh A0, A1, . . . , An twnpoluwnÔmwn f0, . . . , fn enai ousi¸dh (essential). Tìte h arqik morf th torik apalofousa R, w pro thn genik (generic) sunrthsh anÔywsh ω, enai sh me

initω(R) = const ·n∏

i=0

∏

R

cVol(R)i,Fi

,ìpou to deÔtero ginìmeno enai pnw se ìla ta meikt keli R, tÔpou i, th meikt upodiaresh pou epgetai apì thn ω kai ci,Fienai o suntelest toumonwnÔmou me ekjèth to shmeo Fi ∈ Ai. H stajer const enai +1 -1.Gia kje genik sunrthsh anÔywsh ω, h arqik morf initω(R) enaièna akrao mon¸numo th R. To mon¸numo autì antistoiqe sthn koruf toupolutìpou N(R) h opoa èqei upereppedo st rixh (support hyperplane) mekjeto dinusma (normal vector) to ω. An h ω den enai genik sunrthsh,tìte h arqik morf initω(R) antistoiqe se èna polu¸numo to opoo enaiginìmeno apaloifous¸n susthmtwn me sÔnola st rixh , uposÔnola twn Ai(Je¸rhma 4.1 [18). H ω, sth perptwsh aut enai to kjeto dinusma se ènaupereppedo st rixh kpoia èdra tou polutìpou N(R).Enai fanerì pw gia kje genik (generic) sunrthsh anÔywsh ω, upo-logzoume kai mia koruf (ìqi aparathta diaforetik ) tou polutìpou touNeÔtwna th apalofousa , h opoa èqei kjeto dinusma to ω. 'Etsi toje¸rhma 1.2.8 ma parèqei ènan algìrijmo gia ton upologismì tou polutì-pou tou NeÔtwna N(R), th apalofousa : kataskeÔase ìle ti meiktè upodiairèsei tou ajrosmato Minkowski twn polutìpwn tou NeÔtwna twnpoluwnÔmwn tou sust mato kai qrhsimopoi¸nta to je¸rhma 1.2.8, upolì-gise ta akraa mon¸numa th apalefousa . To N(R) enai h kurt j kh twnshmewn aut¸n.Kje koruf tou N(R) pou antistoiqe se èna akrao mon¸numo th R,èqei poll kjeta dianÔsmata ω. Qrhsimopoi¸nta kpoia aut w sunart -sei anÔywsh gia th kataskeu mia meikt upodiaresh , ja upologzoumemèsw tou jewr mato 1.2.8 pnta to dio akrao mon¸numo. Epomènw o pa-rapnw algìrijmo enai anapotelesmatikì afoÔ baszetai sthn apeikìnishtwn meikt¸n upodiairèsewn sta akraa mon¸numa th R, h opoa ìmw den enai1-1. Qrhsimopoi¸nta thn 1-1 kai ep antistoiqa twn diamorf¸sewn meikt¸n19

keli¸n kai twn akrawn monwnÔmwn mporoÔme na tropopoi soume ton algì-rijmo ¸ste na upologzei mìno klsei isodunama meikt¸n upodiairèsewn. Hkataskeu aut anaptÔssetai sto keflaio 4.Klenoume to keflaio autì me èna je¸rhma pou afor sth distash toupolutìpou N(R).Je¸rhma 1.2.9. [18 H distash tou polutìpou tou NeÔtwna N(R) th apalofousa twn poluwnÔmwn fi, i = 0, . . . , n, enai m − 2n − 1, ìpou m =∑ni=0 |Ai|.

20

Keflaio 2Trigwnopoi sei 2.1 Eisagwg 'Estw A, èna peperasmèno sÔnolo shmewn1 ston Rn kai P = conv(A), hkurt j kh tou. 'Ena ploko (simplex) ston R

n enai h kurt j kh enì omoparallhlik anexrthtou (affinely independent) sunìlou, plhjikìthta n+1, p.q. èna eujÔgrammo tm ma ston R, èna trgwno ston R

2, èna tetredroston R3 k.t.l.Orismì 2.1.1. Mia trigwnopohsh T tou A enai mia peperasmènh sullog apì ploka T1, . . . , Tk ⊂ P , ta keli th trigwnopohsh , tètoia pou1. dim(Ti) = n, gia kje i = 1, . . . , k,2. ∪ni=1Ti = P ,3. Gia kje i 6= j, h Ti ∩ Tj enai mia koin èdra (pijan¸ ken ) twn Ti, Tj .Mia trigwnopohsh T tou sunìlou A, enai kanonik (regular), an upr-qei mia genik (generic) sunrthsh anÔywsh (lifting function) ω : A → R,tètoia pou h T na enai h probol twn ìyewn tou ktw (isodÔnama tou nw)peribl mato tou sunìlou A = (a, ω(a))|a ∈ A sto conv(A). To dinusma

w me suntetagmène ta ω(a), kaletai dinusma bar¸n (weight vector) th trigwnopohsh .Parat rhsh 2.1.1. Mia trigwnopohsh enì sunìlou A den enai aparathtona qrhsimopoie ìla ta shmea tou A. Gia pardeigma an A ⊂ R, uprqei miatrigwnopohsh tou A pou apoteletai apì èna mìno ploko, to eujÔgrammotm ma conv(A), kai qrhsimopoie mìno ta shmea pou orzoun ta kra tou. Anmia trigwnopohsh qrhsimopoie ìla ta shmea sto A ja kaletai epikalÔptou-sa (spanning).Mia trigwnopohsh enì polutìpou P , enai mia trigwnopohsh tou sunì-lou twn koruf¸n tou. An to polÔtopo P enai èna n−gwno, tìte o arijmì 1Ta shmea tou sunìlou A den enai aparathta ìla diakekrimèna.21

tn twn trigwnopoi sewn tou enai so me ton (n − 2)−ostì Catalan arijmìtn = Cn−2 =

1

n − 1

(2n − 4n − 2

).To sÔnolo ìlwn twn trigwnopoi sewn enì n−g¸nou qarakthrzetai apìsqèsei pou sundèoun ta mèlh tou. Kje eswterik akm mia trigwnopohsh

Ti tou n−g¸nou, enai mia diag¸nio enì tetrapleÔrou pou sqhmatzetai apìduo trgwna (keli) th Ti. MporoÔme allzonta thn diag¸nio aut me thndeÔterh diag¸nio tou tetrapleÔrou, na kataskeusoume mia trigwnopohshTj pou diafèrei to ligìtero dunatì apì thn Ti. H diadikasa aut kaletaiantistrof akm (edge flip).MporoÔme epomènw na jewr soume ton grfo pou èqei w korufè ti trigwnopoi sei tou n−g¸nou kai w akmè ti antistrofè akm¸n. O gr-fo kaletai grfo antistrof¸n akm¸n (flip graph) twn trigwnopoi sewntou n−g¸nou. Efìson gia kje eswterik diag¸nio tou n−g¸nou uprqeikai mia antistrof akm , èpetai ìti o bajmì tou grfou enai n− 3, dhlad kje trigwnopohsh èqei akrib¸ n−3 antistrofè akm¸n. ApodeiknÔetai eÔ-kola ìti o grfo enai sundedemèno , dhlad apì opoiad pote trigwnopohshuprqei mia akolouja apì antistrofè akm¸n h opoa odhge se opoiad potellh: mia trigwnopohsh tou n−g¸nou kaletai i-tupik (i-th standard) ankje trgwno th èqei koruf thn i. Gia kje llh trigwnopohsh ektì th i-tupik , uprqei mia antistrof akm h opoa auxnei ton bajmì th koru-f i. Se kje trgwno ijk antistrèfoume thn diag¸nio jk, ìpou oi j, k denenai diadoqikè korufè . 'Ena pardeigma fanetai sto sq ma 2.1.

i ii iSq ma 2.1: Antistrofè akm¸n pou odhgoÔn sthn i-tupik trigwnopohsh.O grfo twn trigwnopoi sewn tou n−g¸nou enai èna polÔtopo dista-sh n−3 kai onomzetai associahedron. Enai eidik perptwsh mia kathgora grfwn pou orzontai gia kje sÔnolo shmewn A kai kaloÔntai deutereÔon-ta polÔtopa (secondary polytopes). Ja orsoume austhr ta deutereÔontapolÔtopa sth sunèqeia.Oi trigwnopoi sei enì sunìlou shmewn A sto eppedo èqoun orismène kalè idiìthte pou dieukolÔnoun th melèth tou . Gia pardeigma antistoiqoÔnse èna sundedemèno grfo, to deutereÔon polÔtopo. Gia sÔnola shmewn seq¸rou megalÔterwn diastsewn, akìma kai ston R3, tètoia apotelèsmata denisqÔoun. An ìmw perioristoÔme sto sÔnolo twn kanonik¸n trigwnopoi sewn,mporoÔme na exgoume anloga apotelèsmata.Gia sÔnola shmewn se q¸rou tuqaa distash , h ènnoia th antistro-f akm th diagwnou enì polug¸nou, genikeÔetai se mia antstoiqh ènnoia22

topikoÔ metasqhmatismoÔ pou odhge apì mia trigwnopohsh T se mia llh T ′,¸ste oi duo trigwnopoi sei na diafèroun to ligìtero dunatì.2.2 Kukl¸mataTo katllhlo sunduastikì ergaleo gia na orsoume tou topikoÔ metasqh-matismoÔ mia trigwnopohsh enì sunìlou shmewn A ⊂ Rn enai to kÔklw-ma (circuit). 'Ena kÔklwma Z enai èna elqisto omoparallhlik exarthmèno(minimal affinely dependent) uposÔnolo tou A. Epomènw kje gn sio upo-sÔnolo tou Z enai èna ploko kpoia distash mikrìterh sh tou n.Gia pardeigma èna kÔklwma pl rou distash ston R enai èna sÔnolo tri¸nsuneujeiak¸n shmewn, ston R

2 èna sÔnolo apì tra shmea kai èna pou denbrsketai sti pleurè tou trig¸nou pou orzoun ta upìloipa, ston R3 ènasÔnolo tessrwn shmewn pou sqhmatzoun èna tetredro kai èna shmeo toopoo den an kei se kpoia apì ti ìyei tou k.t.l. (bl. Sq ma 2.2). An tashmea tou A enai se genik jèsh2, tìte èna kÔklwma enai èna uposÔnolotou A plhjikìthta n + 2.

dim(Z)=2

dim(Z)=1

dim(Z)=3

Sq ma 2.2: Kukl¸mata pl rou distash ston R, R2 kai R

3 kai oi antstoi-qe trigwnopoi sei tou .Uprqei mia monadik (mèqri pollaplasiasmoÔ me jetikì pragmatikì arij-mì) omoparallhlik sqèsh metaxÔ twn stoiqewn tou Z :

∑

z∈Z

γzz = 0 µε∑

z∈Z

γz = 0 και γz 6= 0, ∀z ∈ Z.2'Ena sÔnolo shmewn ston Rn enai se genik jèsh an kje n + 1 shmea tou enaiomoparallhlik anexrthta (isodÔnama, an èqoun kurt j kh distash n).23

Prìtash 2.2.1. [9 Kje kÔklwma Z èqei akrib¸ duo trigwnopoi sei T+ = conv(Z \ z) | γz > 0 kai T− = conv(Z \ z) | γz < 0.Epiplèon kai oi duo enai kanonikè .Pardeigma 2.2.1. 'Estw to kÔklwma Z = (0, 0), (1, 3), (2, 2), (3,−1)me antstoiqo dinusma γ = (−1, +4,−5, +2). Oi trigwnopoi sei tou enai oi

T+ = conv(Z \ (1, 3)), conv(Z \ (3,−1)),

T− = conv(Z \ (0, 0)), conv(Z \ (2, 2))ìpw fanetai kai sto sq ma 2.3.1

3

2

1

-1

2 3

+

+

_

_Sq ma 2.3: Oi trigwnopoi sei tou kukl¸mato Z tou paradegmato 2.2.1.Ja qrhsimopoi soume ta kukl¸mata gia na orsoume tou topikoÔ me-tasqhmatismoÔ se mia trigwnopohsh tou sunìlou shmewn A, genikeÔonta thn diadikasa pou akolouj same sti trigwnopoi sei polug¸nwn. H basi-k idèa enai ìti mporoÔme na metasqhmatsoume topik mia trigwnopohsh Ttou A, epilègonta èna katllhlo kÔklwma Z tou A kai antikajist¸nta thntrigwnopohsh tou Z pou epgetai apì thn T , me thn summetrik th . Meton trìpo autì pargoume mia trigwnopohsh T ′ h opoa diafèrei to ligìterodunatì apì th T (bl. sq ma 2.4).O epìmeno orismì qarakthrzei ta kukl¸mata pou ma endiafèroun.Orismì 2.2.1. [15 'Estw A ⊂ Rn èna sÔnolo shmewn, Z ⊂ A èna kÔ-klwma kai T mia trigwnopohsh tou A. H T ja sthrzetai (supported) sto Zan uprqoun Y1, . . . , Ys, uposÔnola tou A (s ≥ 1) kai mia trigwnopohsh T±tou Z (h opoa ja enai h T+ h T−) tètoia pou:1. Gia kje kel (dhlad ploko mègisth distash ) I th T± kai gia kjekel J th T isqÔei: I ⊂ J ⇒ (∃i ∈ 1, 2, . . . , s : J = Yi ∪ I),2. Gia kje i ∈ 1, 2, . . . , s kai gia kje kel I th T±, isqÔei: Yi∪I ∈ T .'Ena isodÔnamo all pio diaisjhtikì orismì enai o epìmeno .24

Orismì 2.2.2. [9 'Estw A ⊂ Rn èna sÔnolo shmewn, Z ⊂ A èna kÔklwmakai T mia trigwnopohsh tou A. H T ja sthrzetai (supported) sto Z anisqÔoun ta paraktw:1. Den uprqoun korufè aplìkwn th T mèsa sto conv(Z) ektì apì tashmea tou Z,2. To polÔtopo conv(Z) enai ènwsh edr¸n twn aplìkwn th T ,3. An conv(I) kai conv(I ′) enai duo keli mia apì ti duo dunatè tri-gwnopoi sei tou conv(Z), tìte gia kje uposÔnolo Y ⊂ A \ Z, toploko conv(I ∪ Y ) emfanzetai sth T an kai mìno an to conv(I ′ ∪ Y )emfanzetai sth T .An |Z| = n + 2, dhlad h distash tou Z enai n, tìte h sunj kh (3) touOrismoÔ 2.2.2 prokÔptei apì thn (2). H sunj kh (3) enai aparathth sthnperptwsh pou ta shmea tou A den enai se genik jèsh, opìte mporoÔme naèqoume kukl¸mata mikrìterh distash apì n. An gia to A isqÔei h upìjeshgenik jèsh , tìte èna kÔklwma Z enai h ènwsh n to polÔ aplìkwn ta opoatèmnontai an dÔo se mia koin tou ìyh. Ta ploka aut an koun kai sthnantstoiqh trigwnopohsh tou A.

(a)

(b)

(c)T

T

T T´

T´

T´

T+

T+

T+

T-

T-

T-

Sq ma 2.4: Topiko metasqhmatismo mèsw enì kukl¸mato Z pl rou di-stash (a),(b) kai mikrìterh distash (c).An T enai mia trigwnopohsh tou A pou sthrzetai se èna kÔklwma Z, tìteh T epgei mia trigwnopohsh tou Z h opoa enai mia apì ti T+, T−. 'Estwìti enai h T+. Ja sumbolzoume me flipZ(T ) th trigwnopohsh pou parnoumeantikajist¸nta ìla ta ploka th morf conv(I ∪ Y ), ìpou conv(I) ∈ T+,25

me ta ploka th morf conv(I ′ ∪ Y ), ìpou conv(I ′) ∈ T− (bl. sq ma 2.4).Profan¸ kai h nèa trigwnopohsh T ′ = flipZ(T ), ikanopoie ti sunj ke touOrismoÔ 2.2.2, epomènw kai aut ja sthrzetai sto Z kai isqÔei flipZ(T ′) =T . Pio austhr h parapnw diadikasa dnetai apì ton epìmeno orismì.Orismì 2.2.3. 'Estw T trigwnopohsh tou A, Z ⊂ A èna kÔklwma stoopoo sthrzetai h T kai Y1, . . . , Ys ìpw ston Orismì 2.2.1. An h T epgeithn trigwnopohsh T+ sto Z, tìte h antistrof akm (bistellar flip) th Tsto Z enai o metasqhmatismì pou orzetai w :flipZ(T ) = (T \ Yi ∪ I|1 ≤ i ≤ s, I ∈ T+) ∪ Yi ∪ I|1 ≤ i ≤ s, I ∈ T−.Orismì 2.2.4. An J ∈ T , enai èna kel th T , ja lème ìti to Z sqetzetaime to (involves) J an J 6∈ flipZ(T ).An to Z sqetzetai me to J , tìte o flipZ katastrèfei to J kai dnei takeli th flipZ(T ) pou den uprqoun sthn T . H efarmog tou flipZ sthntrigwnopohsh T tou A ja ma d¸sei mia nèa trigwnopohsh tou A. 'Omw anh T enai kanonik (regular), den isqÔei pnta ìti kai h flipZ(T ) enai kanonik .'Etsi met apì kje efarmog tou telest flipZ prèpei na elègqoume thnkanonikìthta th trigwnopohsh .2.3 DeutereÔon PolÔtopoT¸ra mporoÔme na orsoume to deutereÔon polÔtopo enì peperasmènou su-nìlou shmewn A ston R

n.Se kje trigwnopohsh T tou A antistoiqoÔme èna dinusma φT , to di-nusma ìgkwn (volume vector), pou orzetai w :φT = (ϕ1, . . . , ϕ|A|), ϕi =

∑

σ∈T , vi∈σ

Vol(σ), (2.1)ìpou to ϕi isoÔtai me to jroisma twn ìgkwn ìlwn twn aplìkwn mègisth distash σ th T pou èqoun to vi w koruf .Parat rhsh 2.3.1. Apì ton prohgoÔmeno orismì prokÔptei ìti ∑|A|i=1 ϕi =

(n + 1)Vol(P ), ìpou P = conv(A).Orismì 2.3.1. To deutereÔon polÔtopo (secondary polytope) Σ(A), tousunìlou shmewn A ⊂ Rn, enai to kurtì perblhma3 twn dianusmtwn ìgkwn

φT , ìlwn twn trigwnopoi sewn tou A.3Ston q¸ro RA twn pragmatik¸n sunart sewn me pedo orismoÔ to A.

26

Parat rhsh 2.3.2. O parapnw orismì tou deutereÔonto polutìpou Σ(A)dìjhke apì tou Gel’fand, Kapranov kai Zelevinskii kai gia kje trigwno-pohsh T tou A dnei ti suntetagmène th antstoiqh koruf vT tou Σ(A)mèsw twn dianusmtwn ìgkwn. 'Ena diaforetikì all isodÔnamo orismì dìjhke apì tou Billera kai Sturmfels sto [2 kai perigrfei to polÔtopo Σ(A)w olokl rwma twn stoibdwn (fibers) th probol π : ∆A → conv(A), ìpou∆A enai èna ploko me |A| korufè kai h π enai h 1-1 kai ep apeikìnish twnkoruf¸n tou ∆A sto A.To sÔnolo twn kanonik¸n trigwnopoi sewn enì sunìlou shmewn A ⊂ R

nsundèetai mèsw mia 1-1 kai ep antistoiqa me ti korufè tou deutereÔonto polutìpou Σ(A). Sqetik èqoume to epìmeno Je¸rhma:Je¸rhma 2.3.1. [9 Gia kje sÔnolo shmewn A uprqei èna polÔtopo Σ(A)tètoio pou:1. H distash tou Σ(A) enai |A| − n − 1.2. Oi korufè tou Σ(A) enai se 1-1 kai ep antistoiqa me ti kanonikè trigwnopoi sei tou A.3. Duo korufè k1, k2 tou Σ(A) me antstoiqe trigwnopoi sei T1 kai T2,sundèontai me mia akm an kai mìno an uprqei èna kÔklwma Z ⊂ A stoopoo sthrzetai h T1 kai T2 = flipZ(T1).Ta dianÔsmata ìgkwn mh kanonik¸n trigwnopoi sewn kanonik¸n upo-diairèsewn, antistoiqoÔn se shmea sto eswterikì tou Σ(A) se shmea stoeswterikì kpoia èdra tou.

Sq ma 2.5: DeutereÔonta polÔtopa enì pentag¸nou kai enì tetrapleÔrou.27

Keflaio 3Aparjmhsh twn kanonik¸ntrigwnopoi sewn3.1 Tèqnasma Cayley'Opw anafèretai sto keflaio 1, o upologismì tou polutìpou tou NeÔtwnath apalofousa enì sust mato poluwnÔmwn f0, . . . , fn ∈ C[x1, . . . , xn],me sÔnola st rixh A0, A1, . . . , An ⊂ Zn, angetai ston upologismì ìlwntwn kanonik¸n lept¸n meikt¸n upodiairèsewn tou ajrosmato Minkowski

P = P0 + · · · + Pn, ìpou Pi = conv(Ai).Mèsw mia antistoqish , gnwst w tèqnasma Cayley (Cayley trick), oupologismì ìlwn twn kanonik¸n lept¸n meikt¸n upodiairèsewn tou P ange-tai ston upologismì ìlwn twn kanonik¸n trigwnopoi sewn enì nèou sunìlouC.Orismì 3.1.1. H emfÔteush Cayley κ, twn polutìpwn P0, . . . , Pn, orzetaiw

C := κ (P0, P1, . . . , Pn) = conv

(n⋃

i=0

(Pi × ei)

)⊂ R

2n+1,ìpou ta ei enai mia omoparallhlik bsh tou Rn. H distash tou sunìlou

κ (P0, P1, . . . , Pn) enai 2n.Gia na kataskeusoume to sÔnolo C, mporoÔme na qrhsimopoi soume w omoparallhlik bsh tou Rn ta dianÔsmata e0 = (0, . . . , 0), e1 = (1, 0, . . . , 0),

e2 = (0, 1, 0, . . . , 0), . . . , en = (0, . . . , 0, 1) ∈ Rn, opìte C ⊂ R

2n ta dianÔ-smata ei, i = 0, . . . , n th orjokanonik bsh tou Rn+1, opìte C ⊂ R

2n+1.H emfÔteush Cayley mpore na oriste genikìtera gia sÔnola shmewnA1, . . . , Ak ⊂ R

n, w ex C := κ (A1, . . . , Ak) =

k⋃

i=0

(Ai × ei) ⊂ Rn × R

k−128

To epìmeno je¸rhma egkajist ènan isomorfismì metaxÔ twn merik dia-tetagmènwn sunìlwn twn poluedrik¸n upodiairèsewn tou C kai twn meikt¸nupodiairèsewn tou sunìlou P .Je¸rhma 3.1.1. (Tèqnasma Cayley)[9 Uprqei mia èna pro èna kai epantistoiqa metaxÔ twn meikt¸n upodiairèsewn tou ajrosmato MinkowskiP = P0 + · · · + Pn kai twn poluedrik¸n upodiairèsewn tou polutìpou C =κ(P0, . . . , Pn). H antistoiqa aut periorzetai se mia antistoiqa twn kano-nik¸n lept¸n meikt¸n upodiairèsewn tou P me ti kanonikè trigwnopoi sei tou C.H idèa tou teqnsmato Cayley enai ìti parnoume èna ploko ston R

nkai topojetoÔme kje Pi pnw apì kje koruf tou, se èna q¸ro orjog¸niopro ton q¸ro ston opoo an kei to ploko, ìpw fanetai kai sto sq ma 3.1gia ti peript¸sei n = 1, 2. Me ton trìpo autì oi eikìne twn Pi brskontaise parllhla upereppeda se ènan q¸ro distash 2n.P e1 1x

P e0 0x

n=1

e1

e0

P e0 0x

P e1 1xP e2 2x

e0

e1e2

n=2

Sq ma 3.1: H idèa tou teqnsmato Cayley gia n = 1, 2.An èqoume mia kanonik trigwnopohsh T tou sunìlou C, mporoÔme nabroÔme thn antstoiqh kanonik lept meikt upodiaresh tou ajrosmato Minkowski P , tèmnonta to C me ton omoparallhlikì upìqwro R

n×∑ 1

n+1ei me opoiod pote llo upereppedo tou R2n pou den enai upereppedo st ri-xh kpoiou apì ta k(Pi). Akribèstera, sth pr¸th perptwsh brskoume miakanonik lept meikt upodiaresh tou upì klmaka ajrosmato Minkowski

1n+1

∑Pi [11, 17.Ta uposÔnola th trigwnopohsh T tou C pou antistoiqoÔn se kelith antstoiqh meikt upodiaresh S tou P , enai sÔmfwna me to epìmenoL mma, ta ploka pl rou distash . 'Ena tètoio ploko prèpei na perièqeitoulqiston èna shmeo apì kje sÔnolo Pi × ei gia na tèmnetai apì tonomoparallhlikì upìqwro R

n × ∑ 1

n+1ei.L mma 3.1.2. [11, L mma 3.2 To R enai kel distash n mia meikt upodiaresh S, tou upì klmaka ajrosmato Minkowski∑n

i=01

n+1Pi an kaimìno an enai h tom enì aplìkou distash 2n mia trigwnopohsh T touC, to opoo perièqei toulqiston èna shmeo (aj , ei) apì kje sÔnolo Ai × ei.To sq ma 3.2 apotele èna pardeigma efarmog tou teqnsmato Cayleygia èna tetrgwno kai èna trgwno. 29

g f

e

d c

ba

ê(a,b,c,d,e,f,g)

(d,0)

(g,1)

(c,0)

(b,0)(a,0

(f,1)

(e,1)

Sq ma 3.2: H efarmog tou teqnsmato Cayley sta sÔnola a, b, c, d kaie, f, g.To tèqnasma Cayley ermhneÔetai algebrik w ex : jewroÔme ti n + 1nèe metablhtè y0, . . . , yn kai kataskeuzoume to polu¸numo

f = y0f0 + y1f1 + · · · + ynfn.To sÔnolo st rixh tou f enai toA :=

n⋃

i=0

Pi × ei ⊂ Z2n+1.H distash tou sunìlou autoÔ enai 2n. Jètoume z = (x, y), N = 2n+1, m =

|A0| + · · · + |An| kai jewroÔme thn A-diakrnousa DA [9. Tìte isqÔei toepìmeno.Prìtash 3.1.3. [9, Prìtash 1.3.1 H torik apalofousa R twn poluwnÔ-mwn f0, . . . , fn isoÔtai me thn A-diakrnousa ∆A.3.2 O algìrijmo aparjmhsh To tèqnasma Cayley ma epitrèpei na upologsoume to polÔtopo tou NeÔtw-na th apalofousa aparijm¸nta ìle ti kanonikè trigwnopoi sei enì nèou sunìlou. Gia to skopì autì uprqoun arketè ulopoi sei ìpw oiPUNTOS [5, TOPCOM [16 k.a. QrhsimopoioÔme ènan apotelesmatikì al-gìrijmo aparjmhsh twn kanonik¸n trigwnopoi sewn enì sunìlou shmewnC pou ofeletai stou T. Masada, H. Imai, K. Imai [14,[12.30

O algìrijmo autì qrhsimopoie mia teqnik aparjmhsh pou kaletai an-tstrofh anaz thsh (reverse search) kai protjhke apì tou Avis kai Fukuda[1.3.2.1 O algìrijmo th antstrofh anaz thsh H antstrofh anaz thsh enai mia teqnik aparjmhsh (kataskeu ) sunìlwnmeglou pl jou diakrit¸n antikeimènwn h opoa mei¸nei thn katanlwsh mn -mh . Sth genik th morf , mpore na jewrhje w h exereÔnhsh enì dèntrouepikluyh , tou dèntrou antstrofh anaz thsh , enì grfou G = (V, E)tou opoou ti korufè jèloume na aparijm soume. Oi akmè tou grfou ka-jorzontai apì mia sqèsh geitnash Adj. Oi akmè tou dèntrou antstrofh anaz thsh , kajorzontai apì mia bohjhtik sunrthsh. H sunrthsh aut mpore na jewrhje w mia topik sunrthsh anaz thsh f se èna prìblhmabeltistopohsh pou orzetai sto sÔnolo twn koruf¸n tou G. Uprqei ètsimia koruf v∗ tou G h opoa enai bèltisth. Diadoqikè efarmogè th su-nrthsh f se kje llh koruf v, pargoun èna monadikì monopti apì thnv sth v∗.IsodÔnama, oi akmè tou dèntrou antstrofh anaz thsh kajorzontaiorzonta mia olik sqèsh ditaxh < twn koruf¸n tou grfou. Gia kjekoruf v, epilègoume apì ìle ti geitonikè th , thn koruf v′ pou enai<-mègisth. 'Etsi pargetai èna monopti apì th v sth v∗.To sÔnolo ìlwn aut¸n twn monopati¸n orzei kai to dèntro antstrofh anaz thsh .H sunrthsh f (h ditaxh <) orzei sto sÔnolo twn koruf¸n tou G miasqèsh gonèa - paidioÔ (epiplèon th sqèsh geitnash ), h opoa belti¸neith qwrik poluplokìthta sugkritik me lle mejìdou aparjmhsh ìpw hanaz thsh kat bjo kai kat plto (Depth-search-first (DFS), Breadth-search-first (BFS)).H antstrofh anaz thsh xekin apì th bèltisth koruf v∗ kai kat thndirkei th mìno oi akmè tou dèntrou antstrofh anaz thsh diatrèqon-tai. Kje for pou aparijmetai mia koruf , aut dnetai sthn èxodo kai denapojhkeÔetai giat h koruf aut mpore na anakthje mèsw th f (mèsw th <) kai epiplèon mporoÔme na ftsoume se aut mèsw enì mìno monopatioÔ.Autì èqei shmantik ofèlh sth qwrik poluplokìthta tou algìrijmou.Gia thn efarmog th antstrofh anaz thsh , prèpei o grfo ti koru-fè tou opoou ja aparijm soume, na enai sundedemèno kai na gnwrzoume ènanw frgma δ sto mègisto bajmì twn koruf¸n tou. H sqèsh geitnash prèpeina pargei seiriak ìlou tou getone kje koruf qwr epanal yei .O epìmeno orismì kajorzei pìte mporoÔme na efarmìsoume thn ant-strofh anaz thsh.Orismì 3.2.1. To (S, d, Adj, f) enai mia dom antstrofh anaz thsh anikanopoie ta epìmena: 31

1. To S enai èna peperasmèno sÔnolo kai to δ enai èna fusikì arijmì ,2. Adj : S × 1, . . . , δ → S ∪ 03. Gia kje koruf v tou grfou kai gia kje i = 1, . . . , δ, h Adj(v, i)enai mia geitonik koruf th v to mhdèn,4. An gia kpoia koruf v kai i, j ≤ d isqÔei Adj(v, i) = Adj(v, j) 6= 0,tìte i = j,5. Gia kje koruf v, to Adj(v, i)|Adj(v, i) 6= 0, 1 ≤ i ≤ δ enai tosÔnolo ìlwn twn geitonik¸n koruf¸n th v,6. H f : S → S enai h sunrthsh pou dnei ton gonèa mia koruf ,f(v) = v f(v) = Adj(v, i), gia kpoio i,7. Uprqei mia monadik koruf r (h rza tou dèntrou), tètoia pou f(r) =r. Gia kje llh koruf v, uprqei èna fusikì arijmì k tètoio pouf (k)(v) = r.Ta pleonekt mata th mejìdou antstrofh anaz thsh enai ta ex : (i)enai euasjhth exìdou (output sensitive), epomènw h qronik poluplokìthtth enai anlogh tou megèjou th exìdou ep èna polu¸numo sto mègejo th eisìdou kai (ii) h qwrik poluplokìtht th enai poluwnumik w pro to mègejo th eisìdou.Je¸rhma 3.2.1. [1 An ikanopoioÔntai oi parapnw proupojèsei , tìte hqronik poluplokìthta th antstrofh anaz thsh enai O(δ · (time(Adj)

+time(f)) · |S|), ìpou time(Adj), time(f) enai oi qronikè poluplokìthte th sqèsh Adj kai th sunrthsh f antstoiqa. H mn mh pou apaitetaienai diplsia tou megèjou enì antikeimènou tou sunìlou S.a

b d

c

r

a

b d

c

r

a

b d

c

r

(i) (iii)(ii)Sq ma 3.3: Pardeigma efarmog th antstrofh anaz thsh sto sÔnoloshmewn S = a, b, c, d, r: (i) o grfo geitnash twn shmewn tou S, (ii)oi sqèsei gonèa - paidioÔ pou orzei h sunrthsh anaz thsh f kai (iii) todèntro antstrofh anaz thsh .32

3.2.2 Perigraf tou algorjmou aparjmhsh twn ka-nonik¸n trigwnopoi sewnJa perigryoume ta basik shmea tou algìrijmou aparjmhsh twn kanonik¸ntrigwnopoi sewn me qr sh antstrofh anaz thsh pou dnetai sta [14, 12.O algìrijmo pou perigrfetai sto [12, qrhsimopoie mia parallag th basik dom antstrofh anaz thsh uiojet¸nta mia olik sqèsh ditaxh sto sÔnolo S.Orismì 3.2.2. To (S, d, Adj, <) enai mia dom antstrofh anaz thsh meolik ditaxh an ikanopoie ta epìmena:1. To (S, d, Adj) ikanopoie ti sunj ke (1) - (5) tou orismoÔ 3.2.1,2. h < enai mia olik sqèsh ditaxh sto S,3. To monadikì stoiqeo r ∈ S pou ikanopoie thnmax< (v ∈ S|∃i v = Adj(r, i) ∪ r) = r, enai to mègisto w pro thn < stoiqeo tou S.'Estw ta sÔnola st rixh A0, A1, . . . , An ⊂ R

n kai C = v1, . . . , vm,ìpou m = |A0| + . . . + |An|, to sÔnolo pou parnoume me thn efarmog touteqnsmato Cayley. H distash tou sunìlou C enai d := 2n.Se kje trigwnopohsh T tou C antistoiqe èna dinusma ìgkwn φT =(ϕ1, . . . , ϕm), ìpou ϕi =

∑∈∆, vi∈σ Vol(σ) (bl. sqèsh (2.1), σ.26). Diats-soume olik ta dianÔsmata ìgkwn ìlwn twn trigwnopoi sewn tou C qrhsi-mopoi¸nta th lexikografik ditaxh1. To deutereÔon polÔtopo Σ(C) tou Cenai to kurtì perblhma twn dianusmtwn aut¸n (bl. enìthta 2.3).Oi akmè tou Σ(C) antistoiqoÔn se antistrofè akm¸n pnw se kpoiokÔklwma tou C. An isqÔei h upìjesh genik jèsh , tìte ta kukl¸mataenai sÔnola apì d + 2 shmea pou sqhmatzoun ploka ta opoa tèmnontai semia koin tou ìyh. ParathroÔme ìti ìla ta kukl¸mata èqoun kurt j khdistash d.To epìmeno L mma prokÔptei apì to upper bound theorem (McMullen).L mma 3.2.2. [12, L mma 10 An jewr soume stajer th distash d, tìteo arijmì twn aplìkwn distash d kai o arijmì twn aplìkwn kje distash se mia trigwnopohsh T tou C, enai sd = O(m⌊ d+1

2 ⌋) kai s = O(m⌊ d+1

2 ⌋),antstoiqa.H dom antstrofh anaz thsh gia to prìblhma th aparjmhsh ìlwntwn kanonik¸n trigwnopoi sewn tou sunìlou C enai h (R, δ, Adj, <), ìpouR enai to sÔnolo ìlwn twn kanonik¸n trigwnopoi sewn tou C, δ = O(ds)enai o mègisto arijmì geitonik¸n trigwnopoi sewn (isodÔnama, o mègisto arijmì kuklwmtwn sta opoa mpore na sthrzetai mia trigwnopohsh, bl.1'Ena dinusma (x1, . . . , xm) enai mikrìtero lexikografik apì èna dinusma(y1, . . . , ym), an gia kpoio i, 1 ≤ i ≤ m isqÔei xi < yi kai xj = yj gia kje j < i.33

paraktw), Adj(T , i) =(h i-ost kanonik trigwnopohsh h opoa mpore naparaqje apì thn T me antistrof akm se kpoio kÔklwma) kai < h olik ditaxh twn trigwnopoi sewn pou epgei h lexikografik ditaxh twn ant-stoiqwn dianusmtwn ìgkwn.Ta aparathta stoiqea mia trigwnopohsh anaparstantai apì ti para-ktw domè :• 'Aploka. Ta shmea tou C enai se genik jèsh. Kje ploko (kel)mia trigwnopohsh T tou C, enai èna sÔnolo apì d + 1 shmea kaianaparstatai jewrhtik apì to dinusma twn deikt¸n twn koruf¸ntou. Kje trigwnopohsh anaparstatai apì èna grfo me kìmbou taplok th kai akmè ti koinè tou ìyei . Oi koinè ìyei twn aplìkwnanaparistoÔn ti sqèsei prìsptwsh metaxÔ tou .Kje koin tètoia ìyh enai èna ploko d shmewn kai thn anaparistoÔmeme ta duo shmea twn duo temnìmenwn aplìkwn pou den an koun sthnìyh aut . H dom aut qreizetai O(dsd) q¸ro, ìpou sd = O(m⌊ d+1

2 ⌋)enai o mègisto arijmì aplìkwn distash d se mia trigwnopohsh touC.Ta shmea tou C den enai se genik jèsh. Sth perptwsh aut prèpeina diathroÔme ta ploka kje distash . Autì gnetai diathr¸nta gia kje trigwnopohsh, to merik¸ diatetagmèno (w pro thn sqèsh⊆) sÔnolo twn edr¸n th (face poset). O q¸ro pou apaitetai enaiO(ds), ìpou s enai o mègisto arijmì aplìkwn kje distash sthntrigwnopohsh (to mègejo tou face poset).

• Kukl¸mata. Ta shmea tou C enai se genik jèsh. DiathroÔme ìlata kukl¸mata pou ikanopoioÔn ti sunj ke (1) kai (2) tou OrismoÔ2.2.2. Apì thn upìjesh genik jèsh èpetai ìti èna kÔklwma enaièna sÔnolo apì d + 2 shmea, ta opoa jewroÔme diatetagmèna w pro tou dekte tou . To sÔnolo ìlwn twn kuklwmtwn anaparstataijewrhtik apì mia lsta twn parapnw sunìlwn shmewn diatetagmènhlexikografik. H kurt j kh enì kukl¸mato apoteletai apì to polÔd+1 ploka kai praktik anaparistoÔme èna kÔklwma me ta ploka aut.O arijmì twn kuklwmtwn enai O(dsd) kai h parapnw anaparstashtou apaite O(dsd) q¸ro.Ta shmea tou C den enai se genik jèsh. Gia kje trigwnopohshT tou C diathroÔme ìla ta kukl¸mata me distash (th kurt tou j kh ) k, gia 1 ≤ k ≤ d, sta opoa sthrzetai h T . Sthn perptwshaut ta kukl¸mata pou diathroÔme prèpei na ikanopoioÔn epiplèon kaith sunj kh (3) tou OrismoÔ 2.2.2. 'Ena kÔklwma distash k enaièna sÔnolo apì k + 2 shmea kai h kurt j kh tou apoteletai apì topolÔ k + 1 ploka distash k. AnaparistoÔme kje tètoio kÔklwma34

mèsw twn aplìkwn aut¸n. O arijmì twn kuklwmtwn enai O(ds) kaih parapnw anaparstash tou apaite O(ds) q¸ro.• DianÔsmata ìgkwn. Gia kje trigwnopohsh T tou C, diathroÔmekai to antstoiqo dinusma ìgkwn φT .Met apì kje efarmog mia antistrof akm prèpei na gnetai enhmè-rwsh twn dom¸n aut¸n. Autì gnetai analutik w ex :• 'Aploka. Ta shmea tou C enai se genik jèsh. Qreizetai na èqoumeto merik¸ diatetagmèno sÔnolo mìno twn d kai d− 1 aplìkwn kai autìgnetai se qrìno O(dsd).Ta shmea tou C den enai se genik jèsh. Prèpei na enhmer¸noumeto merik¸ diatetagmèno sÔnolo twn edr¸n mia trigwnopohsh . Autìgnetai se qrìno O(ds).• Kukl¸mata. Ta shmea tou C enai se genik jèsh. To polÔ d2 ku-kl¸mata diagrfontai prostjentai sth lsta, duo kukl¸mata sugkr-nontai w pro th lexikografik ditaxh se qrìno O(d) kai oi sunj ke tou OrismoÔ 2.2.2 elègqontai se O(d2) gia kje kÔklwma elègqonta tou getone tou. Epomènw h ananèwsh th dom pou anaparist takukl¸mata gnetai se qrìno O(d2sd).Ta shmea tou C den enai se genik jèsh. Se kje antistrof akm ,to polÔ (d+1)s kukl¸mata pou apoteloÔntai apì perissìtera tou enì ploka diagrfontai prostjentai sth lsta. O upologismì twnkuklwmtwn aut¸n gnetai se qrìno O(d4s). O èlegqo twn sunjhk¸ntou OrismoÔ 2.2.2 se O(ds) gia kje kÔklwma. Uprqoun to polÔ mkukl¸mata pou apoteloÔntai apì èna ploko kai o upologismì tou gnetai se qrìno O(d4sm). Epomènw h dom gia ta kukl¸mata mporena ananewje se qrìno O(d4s2).• DianÔsmata ìgkwn. O upologismì tou ìgkou enì aplìkou g-netai se qrìno O(d3), epomènw o upologismì tou dianÔsmato ìgkwnmia trigwnopohsh gnetai se qrìno O(d3s).H trigwnopohsh pou upologzetai me efarmog mia antistrof akm mpore na mhn enai kanonik ra den ja antistoiqe se nèa koruf tou deute-reÔonto polutìpou. Epomènw prèpei na gnetai èlegqo kanonikìthta giakje nèa trigwnopohsh. Autì mpore na gnei me eplush enì probl mato grammikoÔ programmatismoÔ se qrìno O(LP (m−d−1, s)) an den isqÔei h upì-jesh genik jèsh (Prìtash 15, [12) kai se qrìno (O(LP (m− d− 1, dsd))an isqÔei (Je¸rhma 3, [14), ìpou me LP (p, q) sumbolzoume to qrìno eplu-sh enì probl mato grammikoÔ programmatismoÔ q austhr¸n anis¸sewn se

p metablhtè .O algìrijmo qreizetai mia arqik trigwnopohsh gia na xekin sei. Epi-lègoume thn trigwnopohsh me to megalÔtero lexikografik dinusma ìgkwn.35

Aut mpore na upologiste apì opoiad pote llh trigwnopohsh (h opo-a kataskeuzetai qrhsimopoi¸nta katllhlh - genik (generic) sunrthshanÔywsh ) metasqhmatzont thn efarmìzonta diadoqikè antistrofè ak-m¸n pou odhgoÔn se trigwnopoi sei me oloèna megalÔtera dianÔsmata ìgkwn.O qrìno pou qreizetai gia aut th kataskeu enai amelhtèo se sÔgkrishme ton sunolikì qrìno tou algìrijmou aparjmhsh .Sunoyzonta ta parapnw, èqoume ta epìmena jewr mata.Je¸rhma 3.2.3. [14, Je¸rhma 4 An ta shmea tou C enai se genik jèsh, h dom pou orsame epitrèpei thn aparjmhsh twn kanonik¸n trigwno-poi sewn tou sunìlou C me antstrofh anaz thsh se qrìno O(dsdLP (m −d − 1, dsd)|R|) kai q¸ro O(dsd).Je¸rhma 3.2.4. [12, Je¸rhma 13 An ta shmea tou C den enai se genik jèsh, h dom pou orsame epitrèpei thn aparjmhsh twn kanonik¸n trigwno-poi sewn tou sunìlou C me antstrofh anaz thsh se qrìno O(d2s2LP (m −d − 1, s)|R|) kai q¸ro O(ds).

36

Keflaio 4Aparjmhsh twndiamorf¸sewn meikt¸nkeli¸n4.1 Eisagwg Mèsw tou algìrijmou antstrofh anaz thsh tou kefalaou 3 kai tou te-qnsmato Cayley (Je¸rhma 3.1.1), mporoÔme na aparijm soume ìle ti (kanonikè leptè ) meiktè upodiairèsei tou ajrosmato Minkowski P =P0 + . . . + Pn, ìpou Pi = conv(Ai). Epomènw mporoÔme mèsw tou Jewr -mato 1.2.8, na upologsoume ti korufè tou polutìpou tou NeÔtwna N(R)th apalofousa R th ousi¸dou oikogèneia sunìlwn st rixh A0, . . . , An(bl. Orismì 1.2.7).Oi korufè tou polutìpou tou NeÔtwna th apalofousa enai se 1 − 1antistoiqa me ti diamorf¸sei meikt¸n keli¸n, dhlad me ti klsei isoduna-ma twn meikt¸n upodiairèsewn (bl. kai sqìlia met apì to Je¸rhma 1.2.8).Sthn prxh to pl jo twn diamorf¸sewn meikt¸n keli¸n enai polÔ mikrìteroapì to pl jo twn meikt¸n upodiairèsewn tou P . Epomènw , an endiaferì-maste gia ton upologismì tou polutìpou tou NeÔtwna th apalofousa , japrèpei na upologsoume mìno ti diamorf¸sei meikt¸n keli¸n. Autì shma-nei ìti den prèpei na aparijm soume ìle ti kanonikè trigwnopoi sei tousunìlou C = κ(P0, . . . , Pn), all mìno ìse antistoiqoÔn sti diamorf¸sei meikt¸n keli¸n.Gia to skopì autì protenoume thn tropopohsh tou algìrijmou aparj-mhsh twn kanonik¸n trigwnopoi sewn tou kefalaou 3, ¸ste na aparijmekanonikè trigwnopoi sei pou antistoiqoÔn, mèsw tou teqnsmato Cayley,se diamorf¸sei meikt¸n keli¸n. H tropopohsh afor ston entopismì twnkuklwmtwn mia trigwnopohsh T , sta opoa h efarmog antistrof akm ja odhg sei se mia nèa trigwnopohsh T ′, h opoa antistoiqe se nèa diamìr-fwsh meikt¸n keli¸n. 37

JewroÔme ta polu¸numa f0, . . . fn ∈ Cn me antstoiqa polÔtopa tou NeÔ-twna P0, . . . , Pn kai thn eikìna tou C = κ (P0, . . . , Pn) mèsw th emfÔteush

Cayley κ. 'Estw d = 2n h distash tou sunìlou autoÔ. Mia meikt upodia-resh S tou sunìlou P = P0 + . . .+Pn, ja antistoiqe mèsw tou teqnsmato Cayley se kpoia kanonik trigwnopohsh T tou sunìlou C. Efìson h em-fÔteush Cayley enai 1-1 kai ep, ja lème ìti to Z = Z0 + · · · + Zn, Zi ⊂ Pienai èna kÔklwma th S, an to κ(Z) = κ(Z0, . . . , Zn), enai èna kÔklwmath T . H antistrof akm se èna kÔklwma Z ⊂ S, enai h antistrof ak-m sto kÔklwma κ(Z). Gia eukola, an den uprqei knduno sÔgqush , jasumbolzoume ta Z kai κ(Z) me to dio dinusma (Z0, . . . , Zn) kai ta sum-frazìmena ja kajorzoun an anaferìmaste se uposÔnolo th S th T .Tèlo , ja anaparistoÔme kje sÔnolo Zi me to sÔnolo twn koruf¸n tou w Zi = zi,1, . . . , zi,|Zi|, ìpou zj,i ∈ vert(Zi) ⊆ Pi zj,i ∈ vert(Zi) ⊆ Pi × ei,anloga an anaferìmaste sto kÔklwma th S th T antstoiqa.Jumzoume ìti kje kel th trigwnopohsh T enai èna ploko distash d, to opoo sÔmfwna me to L mma 3.1.2, antistoiqe se èna kel distash nth meikt upodiaresh S tou P .Ja lème ìti to kÔklwma Z th T sqetzetai me meiktì kel, an uprqeikel I th T kai meiktì kel R = F0 + · · ·+Fn th S, ¸ste to Z na sqetzetai(Orismì 2.2.4) me to I kai epiplèon I = κ(F0, . . . , Fn), dhlad to I enaieikìna tou R mèsw th emfÔteush Cayley.'Opw kai me ta kukl¸mata, ja sumbolzoume kje meiktì kel R th Skai thn eikìna tou κ(R) th T , me to dio dinusma (F0, . . . , Fn).Zhtme ta kukl¸mata Z th T ta opoa sqetzontai me meikt keli. Hefarmog antistrof akm se èna tètoio kÔklwma ja katastrèfei ta meiktkeli th S me ta opoa sqetzetai kai ja dnei nèa.4.2 H eidik perptwsh th genik jèsh 'Estw ìti gia ta shmea tou C isqÔei h upìjesh genik jèsh , dhlad kjed+1 shmea tou èqoun kurt j kh distash d. Apì thn upìjesh aut èpetaiìti kje kÔklwma mia trigwnopohsh T tou C, enai h kurt j kh distash d, d + 2 omoparallhlik anexrthtwn shmewn, dhlad uprqoun kukl¸matamìno pl rou distash . Kje kÔklwma apoteletai apì to polÔ d ploka,to kajèna distash d. H tom kje zeÔgou apì ta ploka aut, enai miakoin ìyh tou , dhlad èna ploko distash d − 1.JewroÔme to kÔklwma Z th trigwnopohsh T tou sunìlou C. Lìgwth upìjesh genik jèsh kai tou L mmato 3.1.2, to kÔklwma Z enaiènwsh to polÔ d aplìkwn th T , th morf k(Ri) = k(Fi,0, . . . , Fi,n), ìpouPj ⊇ Fi,j 6= ∅ kai Ri = Fi,0 + · · · + Fi,n, gia i = 1, . . . , d, enai èna kel th antstoiqh meikt upodiaresh S.Gia na sqetzetai to Z me èna meiktì kel, ja prèpei toulqiston èna apì38

ta keli Ri na enai meiktì kel th upodiaresh S. Dhlad ja prèpei nauprqoun fusiko p, q, 0 ≤ p ≤ n kai 0 ≤ q ≤ d, ¸ste na isqÔeidim(Fq,i) = 1 για i 6= p και dim(Fq,p) = 0, (Σ)kai epomènw to Rq enai Minkowski jroisma n akm¸n kai mia koruf .Apì ta parapnw prokÔptei ìti gia na aparijme o algìrijmo tou kefa-laou 3, diamorf¸sei meikt¸n keli¸n, prèpei na diathroÔme apì to sÔnolo twnkuklwmtwn mia trigwnopohsh , mìno ta kukl¸mata ta opoa ikanopoioÔn thparapnw sunj kh. Kje kÔklwma anaparstatai apì th lsta twn aplìkwnpou to apoteloÔn. Epomènw elègqonta poia apì aut ta ploka sqetzontaime kpoio meiktì kel kai diagrfonta ta upìloipa, mporoÔme na diathroÔmeèna upersÔnolo twn kuklwmtwn sta opoa h antistrof akm ja d¸sei miatrigwnopohsh pou antistoiqe se nèa diamìrfwsh meikt¸n keli¸n.Parat rhsh 4.2.1. An gia kje kÔklwma, o èlegqo an h trigwnopohsh sth-rzetai se autì (Orismì 2.2.2), prohgetai tou elègqou th sunj kh (Σ),tìte diathroÔme akrib¸ to sÔnolo twn kuklwmtwn, sta opoa h antistrof akm odhge se nèa diamìrfwsh meikt¸n keli¸n. Diaforetik, an o èleg-qo th sunj kh (Σ) prohgetai tou elègqou twn sunjhk¸n tou OrismoÔ2.2.2, tìte diathroÔme èna upersÔnolo twn katllhlwn kuklwmtwn. Autìsumbanei giat mia trigwnopohsh mpore na mhn sthrzetai se kpoia apì takukl¸mata pou ikanopoioÔn th sunj kh (Σ).Apì th upìjesh genik jèsh èpetai ìti gia kje meiktì kel Rj tÔpou

p, me R =∑n

i=0 Fj,i, ìpou Fj,i ⊂ Pi, isqÔei |vert(Fj,i)| = 2 gia kje i 6= p kai|vert(Fj,p)| = 1. Epomènw o èlegqo th sunj kh (Σ) gia èna kÔklwma Zpou apoteletai apì ta ploka k(Rj) = κ(Fj,0, . . . , Fj,n), j ≤ d, sunstataisthn eÔresh toulqiston enì aplìkou gia to opoo ta sÔnola vert(Fj,i)èqoun ti katllhle plhjikìthte .Pardeigma 4.2.1. JewroÔme ìti isqÔei h upìjesh genik jèsh kai n =1. Tìte èqoume ta polu¸numa f0 = c0,1x

a + c0,2xb, f1 = c1,1x

c + c1,2xd ∈

C[x], ìpou oi ci,j enai geniko suntelestè . Ta polÔtopa tou NeÔtwna twnpoluwnÔmwn enai ta eujÔgramma tm mata P0 = [a, b], kai P1 = [c, d]. Apìthn upìjesh genik jèsh , kje polu¸numo prèpei na perièqei akrib¸ dÔomon¸numa. Ta sÔnola P = P0 + P1 kai C = κ(P0, P1) ⊂ R2 fanontai stosq ma 4.1.To C enai èna kÔklwma kai epomènw èqei akrib¸ dÔo trigwnopoi sei

T1, T2 ìpw sto sq ma. Oi antstoiqe meiktè upodiairèsei tou ajrosmato Minkowski P , enai oi S1, S2 oi opoe upologzontai w ex :Gia thn trigwnopohsh T1:

• To kel T1,1 = ((a, 0), (b, 0), (c, 1)) th T1, antistoiqe sto kelR1,1 = (a, b, c) = (a, b) + c, to opoo enai meiktì tÔpou 1.

• To kel T1,2 = ((b, 0), (c, 1), (d, 1)) th T1, antistoiqe sto kelR1,2 = (b, c, d) = b + (c, d), to opoo enai meiktì tÔpou 0.39

Gia thn trigwnopohsh T2:• To kel T2,1 = ((a, 0), (c, 1), (d, 0)) th T2, antistoiqe sto kel

R2,1 = (a, c, d) = a + (c, d), to opoo enai meiktì tÔpou 0.• To kel T2,2 = ((a, 0), (b, 0), (d, 1)) th T2, antistoiqe sto kel

R2,2 = (a, b, d) = (a, b) + d, to opoo enai meiktì tÔpou 1.To kÔklwma C ikanopoie th sunj kh (Σ) giat kje ploko pou an kei seopoiad pote apì ti dÔo trigwnopoi sei tou, sqetzetai me èna meiktì kel th antstoiqh meikt upodiaresh tou P . Epiplèon parathroÔme ìti ta plokatou kukl¸mato C èqoun dianÔsmata plhjikot twn th morf (2, 1) (1, 2)kai epomènw o algìrijmo aparjmhsh ja efarmìsei antistrof akm stokÔklwma autì.'Estw ìti oi ekjète twn poluwnÔmwn f0, f1 èqoun ti timè a = 0, b =3, c = 5, d = 0. Ja upologsoume to polÔtopo th apalofousa N(R) twnpoluwnÔmwn aut¸n qrhsimopoi¸nta to Je¸rhma 1.2.8.Ta dianÔsmata ìgkwn twn trigwnopoi sewn T1 kai T2 tou sunìlou C enaita φT1

= (4, 1.5, 4, 2.5) kai φT2= (1.5, 4, 2.5, 4). To deutereÔon polÔtopoenai to eujÔgrammo tm ma me kra ta shmea aut. Ta meikt keli twnantstoiqwn meikt¸n upodiairèsewn S1, S2 tou sunìlou P = P0 + P1, enai ta

0 + (0, 5), 5 + (0, 3) kai 0 + (0, 3) , 3 + (0, 5).Efarmìzonta to Je¸rhma 1.2.8 upologzoume ta akraa mon¸numa th apalefousa tou sust mato twn dÔo poluwnÔmwn c502c

312, c5

01c311 pou anti-stoiqoÔn sti duo korufè tou polutìpou tou NeÔtwnaN(R): v1 = (5, 0, 3, 0),

v2 = (0, 5, 0, 3). H apalofousa twn f0, f1 enai to polu¸numo R = c502c

312 −

c501c

311.

P0

P1

a b

c d

P +P0 1

b+da+c

S1

a+c b+c b+d

b+da+c a+d

S2

C=ê(P ,P )0 1

T2

T1

(a,0) (b,0)

(c,1) (d,1)

(a,0) (b,0)

(c,1) (d,1)

T1, 2

T1, 1

T2, 2

T2, 1Sq ma 4.1: H perptwsh dÔo poluwnÔmwn se mia metablht ktw apì thnupìjesh genik jèsh . 40

4.3 H genik perptwshSthn enìthta aut anairoÔme thn upìjesh genik jèsh . An gia to sÔnoloC den isqÔei h upìjesh genik jèsh , tìte se mia trigwnopohsh T tou C,mpore na uprqoun kukl¸mata kje distash k = 1, . . . , d. Epomènw giakje tètoio kÔklwma sto opoo sthrzetai h T , prèpei na elègqoume an autìsqetzetai me èna meiktì kel.JewroÔme mia trigwnopohsh T tou C kai èna kÔklwma Z = (Z0, . . . , Zn)distash k ≤ d sto opoo sthrzetai h T . H T epgei mia trigwnopohsh T |Ztou Z h opoa enai mia apì ti T−, T+. 'Estw X1, . . . , Xk ta ploka distash k th T |Z. AfoÔ h T sthrzetai sto Z, apì ton orismì 2.2.2 èpetai ìti kjeploko Xi enai k-èdra kpoiou aplìkou Ui, distash d, th T . Epiplèonuprqoun sÔnola Y1, . . . , Yk ⊂ C \ Z, me kurt j kh distash d − k − 1,¸ste Ui = conv(Xi ∪ Yi). An k = d tìte Yi = ∅ kai Ui = Xi.To Z enai h kurt j kh k + 2 shmewn ta opoa enai omoparallhlikexarthmèna, en¸ kje ploko Xi, (èdra tou Ui) enai h kurt j kh k + 1omoparallhlik anexrthtwn shmewn. Epomènw to kÔklwma Z enai h kurt j kh twn k+1 shmewn enì aplìkou Xi kai enì shmeou c pou an kei sto diok-distato upereppedo me to Xi all ìqi sto sÔnolo twn koruf¸n vert(Ui)tou Ui. 'Etsi èqoume dexei to epìmeno L mma.L mma 4.3.1. An T enai mia trigwnopohsh tou C h opoa sthrzetai seèna kÔklwma Z distash k kai X èna kel th trigwnopohsh T |Z tou Z,tìte uprqei èna ploko U = (U0, . . . , Ur, . . . , Un) th T , to opoo perièqei toX w k-èdra kai to Z grfetai w

Z = (Z0, . . . , Zr ∪ c, . . . , Zn),ìpou Zi ⊆ Ui kai c ∈ Pr × er \ |(Ur).dim(Z)=2dim(Z)=1

U2U1

X1 X2

Y Y

X1

X2

X3

dim(Z)=1

Y

X1X2Sq ma 4.2: H morf ekfulismènwn kuklwmtwn gia d = 2, 3 kai k = 1, 2.ParathroÔme ìti èna kÔklwma Z mpore na grafe me polloÔ diafore-tikoÔ trìpou anloga me to ploko Xi (isodÔnama to ploko Ui) pou jaepilèxoume. 41

Pardeigma 4.3.1. 'Estw to kÔklwma Z = (∅, b, c, d) tou sq mato 4.3.An epilèxoume to ploko X1 = (∅, b, c), to opoo enai èdra tou aplìkouU1 = (a, b, c), tìte to Z grfetai w Z = (∅, b, c ∪ d), d ∈ P1 × e1.An epilèxoume to ploko X2 = (∅, c, d), to opoo enai èdra tou aplìkouU2 = (a, c, d), tìte to Z grfetai w Z = (∅, c, d ∪ b), b ∈ P1 × e1.

dim(Z)=1

U2U1

X1 X1

P e0 0xY=a

P e1 1x

b c dSq ma 4.3: To kÔklwma tou paradegmato 4.3.1.Ta kukl¸mata pou ma endiafèroun enai aut pou sqetzontai me meiktìkel, dhlad ta kukl¸mata ta opoa sqetzontai me èna ploko distash dpou enai eikìna enì meiktoÔ kelioÔ. Apì ton qarakthrismì twn kuklwmtwnpou d¸same parapnw, èpetai ìti èna kÔklwma Z pou sqetzetai me meiktìkel prèpei na perièqei toulqisto èna ploko X to opoo na enai èdra enì aplìkou U = κ(R) th T , ìpou R enai èna meiktì kel th S. H epìmenhprìtash enai mia anagkaa sunj kh gia na sqetzetai èna kÔklwma Z th Tme èna meiktì kel. Epiplèon ma parèqei èna sunduastikì krit rio elègqoutwn kuklwmtwn aut¸n.Prìtash 4.3.2. 1 'Estw Z = (Z0, . . . , Zn) èna kÔklwma mia trigwno-pohsh T tou C kai R = (F0, . . . , Fs, . . . , Fn) èna meiktì kel tÔpou s, th antstoiqh meikt upodiaresh S, me Fi = fi1, fi2 akm gia i 6= s kaiFs = vs, vs ∈ Ps koruf . An to Z sqetzetai me to κ(R), tìte uprqoun0 ≤ r ≤ n, r ∈ N kai c ∈ Pr × er, ¸ste gia ta sÔnola Zi na isqÔei

Zi = Fi × ei η Zi = ∅, αν i 6= r

και (4.1)Zr = Fr × er ∪ c η Zr = fr ∪ c, fr ∈ Fr × er, αν i = r.Apìdeixh Apì to L mma 4.3.1 èpetai ìti uprqoun r ∈ N kai c ∈ Pr × er¸ste to Z na grfetai w

Z = (Z0, . . . , Zr ∪ c, . . . , Zn), Zi ⊆ Fi × ei.AfoÔ to Z enai kÔklwma, uprqei mia monadik (mèqri pollaplasiasmoÔ mejetikì pragmatikì arijmì) omoparallhlik sqèsh metaxÔ twn stoiqewn tou1'H prìtash baszetai sto je¸rhma 5.1 tou [1542

Z: ∑

z∈Z

gzz = 0, µǫ∑

z∈Z

gz = 0 και gz 6= 0, ∀z ∈ Z.Ta shmea z ∈ Z enai th morf (z′, ei), z′ ∈ Pi. Xanagrfonta thnprohgoÔmenh sqèsh qwrzonta tou prosjetèou anloga me to sÔnolo Pisto opoo an koun, èqoumed∑

i=0

∑

z′∈Pi

gz′(z′, ei)

= 0, (4.2)ìpou ∑di=0

(∑z′∈Pi

gz′)

= 0 kai gz′ 6= 0, ∀i ∀z′ ∈ Pi.Ta dianÔsmata ei, ej enai grammik anexrthta epomènw gia kje i 6=j èqoume < ei, ej >= 0 kai < ei, ei >= 1. JewroÔme to dinusma 0 =(0, . . . , 0) ∈ R

n kai sqhmatzoume ta eswterik ginìmena th sqèsh 4.2 mekje dinusma (0, ei), i = 0, . . . , n. Parnoume ti sqèsei :∑

z′∈P0

gz′ = 0, . . . ,∑

z′∈Ps

gz′ = 0, . . . ,∑

z′∈Ps

gz′ = 0, . . . ,∑

z′∈Ps

gz′ = 0, (4.3)dhlad to jroisma twn suntelest¸n gz, twn shmewn z kje sunìlou Zi,enai so me mhdèn. Autì sunepgetai ìti |Zi| 6= 1, ∀i = 0, . . . , n.Telik gia to kÔklwma Z = (Z0, . . . , Zn) èqoume ti paraktw peript¸-sei :1. i 6= r, tìte gia kje Zi isqÔei• |Zi| = 0 ⇐⇒ Zi = ∅ • |Zi| = 2 ⇐⇒ Zi = Fi × ei.2. i = r tìte gia to Zr isqÔei• |Zr| = 2 ⇐⇒ Zi = fi ∪ c, fi ∈ Fi × ei, c ∈ Pr × er • |Zr| = 3 ⇐⇒ Zi = Fi × er ∪ c, c ∈ Pr × r,apì ti opoe prokÔptoun oi sqèsei 4.1. 2Ja kaloÔme ta kukl¸mata me |Zr| = 2, rtia kai ta kukl¸mata me Zr = 3,peritt.ParathroÔme ìti to kÔklwma Z den mpore na perièqei thn koruf (vs, es) ∈

Fs × es, tou tÔpou s meiktoÔ kelioÔ R, par mìno an s = r kai to sÔnolo Zrtou Z enai th morf Zr = Fs × es ∪ c = (vs, es), c.SÔmfwna me thn prìtash 4.3.2, gia na aparijme o algìrijmo tou kefala-ou 3, mìno ti diamorf¸sei meikt¸n keli¸n ja prèpei na ektele antistrofè akm¸n mìno se kukl¸mata ta opoa ikanopoioÔn ti sqèsei 4.1.43

Pardeigma 4.3.2. 'Estw ìti èqoume ta polu¸numa f0 = c0,1xa + c0,2x

b,f1 = c1,1x

c + c1,2xd + c1,3x

e ∈ C[x], ìpou oi ci,j enai geniko suntelestè .An upojèsoume ìti a < b kai c < d < e, tìte ta polÔtopa tou NeÔtwnatwn poluwnÔmwn enai ta eujÔgramma tm mata P0 = [a, b] kai P1 = [c, e].To polu¸numo f1 perièqei tra mon¸numa kai to f0 dÔo. Epomènw gia tosÔnolo C = κ(P0, P1) den isqÔei h upìjesh genik jèsh afoÔ to sÔnoloP1 × e1 perièqei tra suneujeiak shmea. Oi trigwnopoi sei tou sunìlou Cpou antistoiqoÔn se diamorf¸sei meikt¸n keli¸n, kaj¸ kai oi antstoiqe diamorf¸sei meikt¸n keli¸n tou sunìlou S = P0 + P1, fanontai sto sq ma4.4. Oi antistrofè akm¸n kai ta antstoiqa kukl¸mata sta opoa autè ekteloÔntai enai ta paraktw:1. flipZ1

(T1) = T2, ìpou to kÔklwma Z1 sqetzetai me to meiktì kel tÔpou0, R = (F0, F1) = (a, c, e) kai grfetai w :

Z1 = (∅, F1 × 1 ∪ (d, 1)) = (∅, (c, 1), (e, 1) ∪ (d, 1)) .To kÔklwma Z1 enai perittì.2. flipZ2(T2) = T3, ìpou to kÔklwma Z2 sqetzetai me to meiktì keltÔpou 0, R1 = (F1,0, F1,1) = (a, d, e) kai to meiktì kel tÔpou 1,

R2 = (F2,0, F2,1) = (a, b, e). To Z2 enai èna rtio kÔklwma kaigrfetai w :Z2 = (F1,0 × 0 ∪ (b, 0), F1,1 × 1) = ((a, 0), (b, 0), (d, 1), (e, 1)) , Z2 = (F2,0 × 0, F2,1 × 1 ∪ (d, 1)) = ((a, 0), (b, 0), (d, 1), (e, 1)) .3. flipZ3

(T3) = T4, ìpou to kÔklwma Z3 sqetzetai me me to meiktì keltÔpou 0, R1 = (F1,0, F1,1) = (a, c, d) kai to meiktì kel tÔpou 1,R2 = (F2,0, F2,1) = (a, b, d). To Z2 enai èna rtio kÔklwma kaigrfetai w :Z3 = (F1,0 × 0 ∪ (b, 0), F1,1 × 1) = ((a, 0), (b, 0), (c, 1), (d, 1)) , Z3 = (F2,0 × 0, F2,1 × 1 ∪ (c, 1)) = ((a, 0), (b, 0), (c, 1), (d, 1)) .4. flipZ4

(T4) = T5, ìpou to kÔklwma Z4 sqetzetai me me ta meikt ke-li tÔpou 0, R1 = (F1,0, F1,1) = (b, c, d) kai R2 = (F2,0, F2,1) =(b, d, e). To Z4 enai èna perittì kÔklwma kai grfetai w :

Z4 = (∅, F1,1 × 1 ∪ (e, 1)) = (∅, (c, 1), (d, 1), (e, 1)) , Z4 = (∅, F2,1 × 1 ∪ (c, 1)) = (∅, (c, 1), (d, 1), (e, 1)) .44

An jèsoume a = 0, b = 4, c = 5, d = 2, e = 0, parnoume ta polu¸numaf0 = c0,1 + c0,2x

4, f1 = c1,1x5 + c1,2x

2 + c1,3 ∈ C[x].Ja upologsoume to polÔtopo th apalofousa N(R) twn poluwnÔmwn au-t¸n.Ta dianÔsmata ìgkwn twn pènte trigwnopoi sewn tou sunìlou C enai taφT1

= (4.5, 2, 4.5, 0, 2.5), φT2= (4.5, 2, 3.5, 2.5, 1), φT3

= (3, 3.5, 1.5, 4.5, 1),φT4

= (2, 4.5, 1.5, 2.5, 3) kai φT5= (2, 4.5, 2.5, 0, 4.5).Oi antstoiqe meiktè upodiairèsei S1, . . . , S5 tou sunìlou P = P0 +P1,enai oi

S1 = 0 + (0, 5), 5 + (0, 4),S2 = 0 + (0, 2) , 0 + (2, 5), 5 + (0, 4),S3 = 0 + (0, 2) , 2 + (0, 4), 4 + (2, 5),S4 = 0 + (0, 4) , 4 + (0, 2), 4 + (2, 5) kaiS5 = 0 + (0, 4) , 4 + (0, 5).Efarmìzonta to Je¸rhma 1.2.8 upologzoume ta akraa mon¸numa th apalefousa tou sust mato twn dÔo poluwnÔmwn:

c50,1c

41,1, c2

0,1c41,2c

30,2, c2

0,1c30,2c

41,2, c2

0,1c30,1c

41,3, c5

0,2c41,3pou antistoiqoÔn sti trei korufè tou polutìpou tou NeÔtwna N(R): v1 =

(5, 0, 4, 0, 0), v2 = (0, 5, 0, , 0, 4), v3 = (2, 3, 0, 4, 0).Ta akraa mon¸numa pou upologzontai enai pènte, ìse kai oi trigwno-poi sei tou sunìlou C all uprqoun duo ìmoia zeÔgh. 'Etsi to polÔtopotou NeÔtwna th apalofousa enai to trgwno me korufè ta dianÔsmatav1, v2, v3.H apalofousa twn f0, f1 enai to polu¸numoR = c5

02c413 + 2c0,1c

402c

212c

21,3 + c2

0,1c30,2c

41,2 − 4c3

0,1c20,2c

21,1c1,2c1,3 + c5

0,1c41,1.

45

a+c b+c b+e

S5

P0

a b

P1

c

d

e

T1

(a,0) (b,0)

(c,1) (d,1) (e,1)

( (c,1),(d,1),(e,1) ) ,Æ

( (a,0),(b,0) , (d,1),(e,1) )

( (a,0),(b,0) , (c,1),(d,1) )

( , (c,1),(d,1),(e,1) )Æ

b+ea+c a+e

S1

R

a+c a+d a+e b+e

S2

R1 R2

a+c b+da+d b+e

S3

R1 R2

a+c b+db+c b+e

S4

R1R2

(a,0)

T4

(a,0) (b,0)

(c,1) (d,1) (e,1)

T5

(a,0) (b,0)

(c,1) (d,1) (e,1)

T2

(a,0) (b,0)

(c,1) (d,1) (e,1)

T3

(b,0)

(c,1) (d,1) (e,1)

Sq ma 4.4: H genik perptwsh dÔo poluwnÔmwn se mia metablht .46

Keflaio 5Mia efarmog sthnalgebrikopohsh5.1 Eisagwg Sto keflaio autì parousizetai mia efarmog tou algìrijmou aparjmhsh diamorf¸sewn meikt¸n keli¸n sthn algebrikopohsh mia parametrik dosmè-nh kampÔlh epifneia . O algìrijmo algebrikopohsh IPSOS, stonopoo efarmìzontai oi mèjodoi th ergasa aut , èqei anaptuqje apì tou I. Emrh kai H. Kotsirèa [7,[8. O IPSOS qrhsimopoie ton algìrijmo apa-rjmhsh kanonik¸n trigwnopoi sewn PUNTOS tou J. De Loera [5, o opoo baszetai sthn mèjodo kat plto anaz thsh kai èqei ulopoihje se MA-PLE. Efìson o PUNTOS baszetai sthn mèjodo th kat plto anaz thsh ,èqei qwrik poluplokìthta Ω(T ), ìpou T enai to pl jo twn kanonik¸n tri-gwnopoi sewn tou sunìlou shmewn th eisìdou tou. H poluplokìthta aut kajist apagoreutik th qr sh tou se eisìdou metrou èw meglou megè-jou . Protenetai h tropopohsh tou algorjmou IPSOS ¸ste na qrhsimopoieton proteinìmeno algìrijmo aparjmhsh diamorf¸sewn meikt¸n keli¸n kaiexetzetai h dunatìthta efarmog enì algìrijmou probol pou anamènetaina belti¸sei peraitèrw th poluplokìthta tou.5.2 O algìrijmo IPSOSO algìrijmo IPSOS qrhsimopoie ergalea apì thn algebrik gewmetra kaiidiatera apì thn jewra arai apaloif gia na upologsei èna upersÔnolotou sunìlou st rixh th algebrik exswsh mia parametrik domènh upe-repifneia . Parousizoume mia tropopoihmènh ekdoq tou algorjmou pouqrhsimopoie ton algìrijmo aparjmhsh twn diamorf¸sewn meikt¸n keli¸n.'Estw ìti ma dnetai h parametrik èkfrash mia uperepifneia

xi =Pi(t)

Q(t), i = 0, . . . , n.47

JewroÔme ta polu¸numa fi = xiQ(t) − Pi(t) ∈ C[t], t = (t1, . . . , tn), mesÔnola st rixh ta Ai ⊂ Zn kai antstoiqa polÔtopa tou NeÔtwna Pi. Stapolu¸numa fi èqoume antikatast sei tou sugkekrimènou suntelestè twn

Q(t) kai Pi(t) me genikoÔ (generic) suntelestè ci,j . H algebrik exswshth uperepifneia enai h apalofousa tou sust mato poluwnÔmwn fi, methn propìjesh ìti den uprqoun base points kai h parametrikopohsh enai1-1. An den isqÔei h teleutaa sunj kh, tìte h apalofousa tou sust mato isoÔtai me èna pollaplsio th algebrik exswsh .Ta b mata tou algorjmou enai:1. 'Orise ta polu¸numa fi =∑

cijtaij ,∈ C[t], ìpou aij ∈ Ai kai cij geniko(generic) suntelestè .2. KataskeÔase to sÔnolo C efarmìzonta to tèqnasma Cayley sta polÔ-topa tou NeÔtwna P0, . . . , Pn kai efrmose ton tropopoihmèno algìrij-mo aparjmhsh trigwnopoi sewn sto sÔnolo autì. Me ton trìpo autìupologzoume ìle ti diamorf¸sei meikt¸n keli¸n tou ajrosmato

Minkowski P = P0 + . . . + Pn.3. Upolìgise ta akraa mon¸numa (korufè tou polutìpou tou NeÔtwnaN(R), th apalofousa ). To sÔnolo autì maz me ìla ta akèraiashmea sto eswterikì tou N(R), enai èna upersÔnolo tou sunìloust rixh th apalofousa .4. Metètreye to sÔnolo st rixh , apì èna sÔnolo monwnÔmwn th morf ∏

ceij

ij , se èna sÔnolo monwnÔmwn w pro ti metablhtè xi.5. Qrhsimopoi¸nta frgmata ston bajmì th algebrik exswsh , ap-leiye ìsa mon¸numa den ikanopoioÔn ta krit ria.Kat thn ektèlesh tou b mato 2 tou algorjmou, upologzontai ìloi oi me-riko meikto ìgkoi MV−i (bl. Pìrisma 1.2.5) oi opooi kai dnoun ton bajmìth algebrik exswsh w pro kje metablht xi xeqwrist. H plhroforaaut qrhsimopoietai sto b ma 5 gia thn apaloif ìswn monwnÔmwn den èqounkatllhlo sunolikì bajmì.5.3 Algìrijmo probol Ta polu¸numa fi ta opoa orzontai sto b ma 1 tou algìrijmou IPSOS èqoungenikoÔ (generic) suntelestè , oi opooi sto b ma 4 antikajstantai apìsunart sei twn xi. H diadikasa aut mpore na jewrhje w probol toupolutìpou N(R), distash d = m − 2n − 1, m =∑n

i=0 |Ai|, se èna q¸rodistash k = n + 1. Sun jw k = 2, 3 anloga me ton an zhtme thnalgebrik exswsh kampÔlh epifneia .'Estw P , h probol aut tou N(R). Pollè apì ti korufè tou polu-tìpou N(R) pou upologzei o algìrijmo , ja probllontai sto eswterikì48

tou polutìpou P . Oi korufè autè den suneisfèroun sth diadikasa eÔresh th algebrik exswsh kai mia shmantik beltwsh ja tan h tropopohshtou algorjmou aparjmhsh tou polutìpou th apalofousa ¸ste na mhnti upologzei. Autì enai isodÔnamo me thn exereÔnhsh tou polutìpou N(R)kat m ko enì monopatioÔ pou perièqei toulqiston ìle ti korufè oiprobolè twn opown enai korufè tou P .Duo pijanè prosseggsei tou probl mato enai oi akìlouje :• H exereÔnhsh tou polutìpou N(R) gnetai apì ton algìrijmo antstrofh anaz thsh me bsh th taxinomhmènh lsta twn kuklwmtwn mia trigwnopoh-sh . Jèloume na kajorsoume poia enai ta katllhla kukl¸mata, sta opoah efarmog antistrof akm odhge apì mia koruf tou N(R) pou probl-letai se koruf tou R, se mia geitonik th me thn dia idiìthta.• 'Estw ìti to polÔtopo P orzetai apì ti xi = 0, i = 1, . . . , k, dhla-d problloume to polÔtopo N(R) diagrfonta ti pr¸te k suntetagmène x1, x2, . . . , xk. 'Ena upereppedo ja lègetai kjeto an perièqei ti suntatagmè-ne autè . H tom twn kajètwn uperepipèdwn st rixh (support hyperplanes)me to polÔtopo N(R) enai mia ìyh tou polutìpou pou orzetai apì korufè tou pou probllontai sto pergramma tou polutìpou P . O stìqo ma enaina upologsoume autè ti korufè .'Estw ìti èqoume upologsei mia koruf v tou N(R) h opoa problletaise koruf tou polutìpou P . H epìmenh katllhlh koruf tou N(R) upo-logzetai elègqonta gia kje geitonik koruf w th v, an enai akraa w pro ti probolè twn geitonik¸n th koruf¸n.

N(R)

P

49

Eplogo Sth ergasa aut exetsame apotelesmatikoÔ trìpou upologismoÔ tou po-lutìpou tou NeÔtwna th torik apalofousa enì sust mato n + 1 po-luwnÔmwn se n metablhtè me sumbolikoÔ suntelestè .SÔmfwna me to je¸rhma 1.2.8, uprqei mia 1-1 kai ep antistoiqa twndiamorf¸sewn meikt¸n keli¸n kai twn akrawn monwnÔmwn th apalofousa .Epomènw gia ton upologismì tou polutìpou tou NeÔtwna th apalofousa arke o upologismì ìlwn twn diamorf¸sewn meikt¸n keli¸n.To tèqnasma Cayley (je¸rhma 3.1.1) angei ton upologismì ìlwn twnlept¸n kanonik¸n meikt¸n upodiairèsewn tou ajrosmato Minkowski twnpolutìpwn tou NeÔtwna twn poluwnÔmwn tou sust mato , ston upologismììlwn twn kanonik¸n trigwnopoi sewn th eikìna twn polutìpwn aut¸n mèswth emfÔteush Cayley.Tropopoi¸nta ton algìrijmo aparjmhsh kanonik¸n trigwnopoi sewntou [12 mporoÔme, me thn efarmog antistrof¸n akm¸n se katllhla kukl¸-mata ta opoa prosdiorzontai apì thn prìtash 4.3.2, na upologsoume mìnoti kanonikè trigwnopoi sei pou antistoiqoÔn sti diamorf¸sei meikt¸n ke-li¸n.Tèlo d¸same mia efarmog twn mejìdwn aut¸n perigrfonta ènan al-gìrijmo algebrikopohsh .

50

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[8] I.Z. Emiris and I.S. Kotsireas. Implicitization exploiting sparseness. In R. Ja-nardan, M. Smid, and D. Dutta, editors, Geometric and Algorithmic Aspectsof Computer-Aided Design and Manufacturing, volume 67 of DIMACS, pages281–298. AMS/DIMACS, 2005.

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Euret riojroisma Minkowksi, 13ploko, 21IPSOS, 47Laurent polu¸numa, 13akraa mon¸numa, 18antstrofh anaz thsh, 31antistrof akm , 26arqik morf , 19deutereÔon polÔtopo, 26dinusma ìgkwn, 26dinusma bar¸n, 21diamorf¸sei meikt¸n keli¸n, 18emfÔteush Cayley, 28genik jèsh, 23je¸rhma Bezout, 14je¸rhma Bernstein, 15kanonik trigwnopohsh, 21kanonik upodiaresh, 17meikt upodiaresh, 16meiktì ìgko , 14ousi¸dh sÔnola st rixh , 16sunrthsh anÔywsh , 17Tèqnasma Cayley, 29torik apalefousa, 15trigwnopohsh, 21

53