1. Introduction and notations.

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1 Alyi Ar om plii orml or q o ory mr Rol Gro Lyé olyl Roièr, r i lir ill, B Tolo Fr Emil : rolgro@orgr W y hr q o ory mr, o ll h o ory polyomil o gi rm om orhogol or h mr Th mi rl i orml mig plii h iy o y rm o h q, r om hypoh W gi om ppliio lo ri irprio o h Forir oii mlipl igrl roio oio W oir proiliy iy io o irl o wih Th ilj rorm o h mr o iy i i o y h orml: [] W o Hilr o ormli polyomil or h li ir pro :, g / g > g o h oi Hilr p L, < W ll X X h ory polyomil oi wih X L rll h wll ow rl low : [] poii mr o oi o iy io µ hig ij rormio gi y h orml: µ, h ory polyomil orm orhogol mily or h ir pro i y µ W hily ll µ h ory mr oi wih Th mom o orr o hi w mr i gi y w ormli µ, w iro ppr q o proiliy iy io : µ o w oi h pro wih h rig wih h or ry igr, i h ormli ory mr o h

2 W op ow h ollowig oio : ; Th ilj rorm o will imply rpr y A lil hm o oio rio Aorig wih prio oio, w h or ry : ] [ Thi lo wri : Th w oi igrm o oio rio : Aorig o lil rl o h hory [3,] w h h orml :, wih h rlio rig wih : ; Th rmi implii o : 3 W immily h : L y hi ir i h ighorhoo o iiiy i qil o i iy o proiliy i oioly polyomil i wih lig rm ql o Th, i h ighorhoo o iiiy : i qil o 3

3 3 Noig h h gr o i h o i, w ol h h rio F i pproim or o yp [ / ] [] Aorig h li hory, hi rio i imply ql o h qoi o w h h proporioliy : Th o h rrr rlio, h lig oii o i ql o, or ry igr, o w h, wih lig oii o From h orml o : w ow 3 llio o oii Aohr rl o h hory o ilj rorm h i h ighorhoo o iiiy : i qil o, wih, prio i whih i h lig oii o [6] omprig wih h prioly oi qil, w :, o : 3 Th orml implii h o : 3 Rll ow h i h lil rrr rlio o hr rm, wri 33, w h, o w oi 3

4 Epliiio o h iy W rll h irio orml o ilj rro whih llow o ror h iy rom i ilj rorm r om hypoh : lim π i i i A impl o ppliio ppr wh h iy i oio io or omp irl W ppo hr h iiil iy iyig h hypoh o irio w ppo lo h i o lim i i or ll or h irl Thi io i ll h rr o h mr o iy llow o plii h ory mr oi wih : π µ [5] Ur h hypoh, h iy m plii h ilj rro h orml 3 o : A lmry llio h l o : π Now rll h orml : From ; ; ; w ily : o, h orml implii o : π

5 5 i ll h io r proiliy ii, w h l o h ollowig igrl : 3 π om mpl [5,6,7] For h iorm Lg mr or [,] : 3 ] l [ π i h hi ormli Lgr polyomil o orr For or [, [ ] Ei [ π i h Lgrr polyomil o orr Ei h poil igrl For h Gi mr π or h rl i ] ri [ π π i h Hrmi polyomil o orr ri h imgiry rror io For h Thyh mr o h o i π or [-,] hi h q o ormli ory mr i o o w i h lil rlio oig h Thyh polyomil :

6 5 A irprio o h Forir oii W rll h ollowi rl : [5] Th opror g rig ory polyomil o oio lir mp T liig h p L, o h Hilr p L, µ rriio o h hyprpl H o h io orhogol wih oi iomri io or oh orm rpily o, or y opl, g o lm o L,, w h: 5 < / g > < / > < g / > < T / T g > µ B o h ormliio prorm i h q o ory mr, w lo h iomri hrr o h oi rorm w o H h p o lm o L, orhogol wih, T i ow iomri mp liig H, o L, o, or y opl, g o L,, w wri : 5 < / g > < / > < g / > < T / T g > W will ow grli hi orml y ompoig h rorm W iro F T o T o o T o T o F T A lmry rrr l o : F 53 mlipl igrl o orr j j L opl, g o lm o L, wih g w h rom 5 : < / g > < T / / T g > µ < F F g > Am rhr g orhogol wih, ill i L, W h F i orhogol wih i, g / g > / < F F > L, o w g : < g Mor grlly, i g i orhogol ry rm o,,, i L, W h h rlio 5 : < / g > / < F F g > 6

7 7 Thi orml o will llow o m plii h Forir oii o gi io mlipl igrl B,, w irly rom 5: 55 F F > < > < / / Rll ow pr rl 3: W lo ily rl F, lig oii o, ry io i iy o proiliy o h orml 55 implii o : 56 j j W ol hi prgrph y prilr ppliio o hi orml W oir hr h io :, wih i igor or ry opror T piilly, lmry llio gi : T wih By lmry ompoiio, w g : F o, rom 56 : Thi wri : A mor pliily : By impl qoi : o w h or : 57

8 8 Apply hi orml o h hyh mr o h o i or h irl ], [ W h π 8, 8 8, h lig oii o i A i hi h q o ormli ory mr i o, w rom 57 : 8 8 q, wih q q o w h i [,], L q No h q q q o w g : O rogi hr h lil grig io or h Thyh polyomil 6 omplm ohr orml Fir mom o From h orml w g : Th o h proporiliy, hi wri : w rll h lil hr-rm rlio 33 :, w immily h qlii : ; No h h l wo gi : Uig, h rlio i rl i igrl orm y π 6

9 9 Rr o Uig orml 3 w m plii h rr o, i y : lim i i Ar om lmry llio h o h irio orml o ilj rro, w g : π π 6 Orhogol polyomil oi wih From h oplig orml µ w ily pr h o primry U ory V orhogol polyomil o µ hrogh : V U 63 From h iomri hrr o T, w h or µ Wh w ormli µ o µ, w iro U o p h orm ql o B µ T T Th m wy w hg V o V o w g T µ T T U V T µ Th rlio 63 rl i o : ] [ 6 Mor grlly i w o, h o polyomil oi wih, w wri i mri orm : : 65

10 Now iro M M M M Π W ily rom 65 : Π 66 w o Π B A, w oi h rlio : B A B A ppr lr h iy h lil hr-rm rlio or h iiil mr : Aorig o h lil hory, w wri : Th ompo,,, r rmi ig h iiil oiio Th q r wih ; ; By 6 w h ; A ir llio wih mri gi : ; ; ; o w g : ; ; ; oqly : For Π ; ] [ 67

11 Rr [] hrii Brg, Al l Flé i Tolo, ér 6 Vol 5 996, p9-3 [] Jhrm, O h mror o h org o h ilj oi rio, TrAmr Mho ol 35 pp [3] Br, G A Jr Th Thory Appliio o Th Approim Mho A i Thoril hyi, Vol E K A Brr Nw Yor: Ami r, pp -58, 965 [] GA Br, Jr Gr-Morri, é pproim, mrig Uiriy r, Loo 996 [5] R Gro R Ai ri, r 7 Vol 35 pg [6] ANiioro, V Oro, Elm l héori oio péil, Eiio Moo, 983 [7] M Armowi A g, Hoo o Mhmil Fio, Niol Br o r, 96, rpri or liio, 965

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