Po -n -(1+ + akhovanov/ma2_fall/files/lect_5.pdf · 2020. 9. 23. · -2 Prop 1 1dgra doain R is a...

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R- m mn t -R, as usul Tua is a mo n s R ln)=a ieu as nd R + no n Sum 1n R Po ssicies Brn ( r) suu -n -(1+ + (a) p(7¢) >R, hu p = R R orduns7 as sd tranas2, 2L% J, a, R, C, Z(). (6) 6) is o i.cse. ku (e)#o. Sia Z isa PLO, a s s prinde , L (e)= (n) ,no ()> (-n) Ten Tu inas 2 in R sa Aie esius moulo n o, n- =0 inR e : R>72/(a), 7L/n ra um a A, as a R Coadaas e tdhn Z su u dauisans A SSu R i AI K= F P aoe nekm 0 P°n siu QF no F 2Z , en F as Fe- P Fp- o, P RA ueidey m P a Exe F A FL polunomi s tn F, ac (F Cz)) -iSnahard F Rac (F(z1) ads ,), () e (x) Coe de As in F. tequvaara wladan, (e) f o () #o

Transcript of Po -n -(1+ + akhovanov/ma2_fall/files/lect_5.pdf · 2020. 9. 23. · -2 Prop 1 1dgra doain R is a...

  • R- m mn t -R, as usul

    Tua is a mo n s R ln)=a ieu as nd R

    + no n

    Sum 1n R

    Po ssicies Brn ( r) suu -n -(1+ +

    (a) p(7¢) >R, hu p =

    R R orduns7 as sd tranas2, 2L% J, a, R, C, Z().

    (6) 6) is o i.cse. ku (e)#o. Sia Z isa PLO, a

    s s prinde , L (e)= (n) ,no ()> (-n)

    Ten

    Tu inas 2 in R sa Aie esius moulo n

    o, n- =0 inR e : R>72/(a),

    7L/n ra um a A, as a R Coadaas e tdhn Z

    su u dauisans

    A SSu R i AI K= F

    P aoe nekm 0 P°n

    siu QF no F 2Z , en F as

    Fe- P Fp- o, P RA ueidey m P a

    Exe F A FL polunomi s tn F, ac (F Cz)) -iSnahard F Rac (F(z1) ads ,), () e (x) Coe de As in F.

    tequvaara wladan, (e) f o () #o

  • -2 Prop 1 1dgra doain R is a sus RAF, tluu

    aO mo monpm ac (R) F L xtnds inus n R F

    rac R SudhB's u 1

    B? Ele menks k oc (R) a pa1s (, ) , f,

    abeR B(ab)=Jl) J(L)

    xei: )B Sa o momep sn,

    2 s R

    an

    Prope sidos Sans wk ar1ncson

    axdsto an neusu. u h Saehons Rrac (R)= 8(R)

    F.

    Q= rac (Z) tx e F eE cF

    TF F åt ains suldl 2

    Pi RASs.

    FF, sa ta clanacktusic isP ch (F=P

    h C)Fo a dhannchiis be of F is O c F, so

    Rer xezu ) ?lr loma R

    is a s a kll , Re

    T Smeu dR Al dains R is isomophie &(R)>Yae (R

    b)o cld Anks : eT inakU t.

  • -3 F-sd, Flx]

    Savisan Jlad, Gne polyoil s (a)q(x)e Fl*] und tm oynomi als

    &(oc) ()am

    Tonsiued q (e),s(x) e dvid (x) (x) m a uwain le

    Snsuin on

    cz) < g(*) (n ca) don c) ( x)* {(x) (e)=lx)

    nm &( a**o,-,"*. a o

    9 ( 6."*. *

    ) = a, ba(b.m x"6z '4 +b). =a,**o, b C + Can inet in F, mtos tta Psa iA, 6en o i cln tnws

    -m

    , ta, -*" T. «a-(a,c * +a, b5

    - a, 6. b., ) 2

    )-a8**)¢ (z) as dau n Proco S adlucto

    ,z) as a

    ()q, (()+, (*) )=q (x)a*)ar, (x),12) (2) +, (*) qx)q(x) +2(x) (-2)a(e) = , («) -, (*)

    wless 2, LtS

  • Diso pono i d's -1

    a ema

    o,,2} 2+2 |

    8(e)=2x+x2x +* () +22 -1 2= - mos

    ensen 2ct

    +0x +2x+x S - -

    2 + 2xt x

    2 -2x- 2* =mals)

    2t

    2x*x = (*2x) (xx -t) 2oc+

    nenie (L ePhiat is 1)Can di g(x) e)u wen oun RC«, R a is monde, rud dop coefhao

    htnu thl in R

    7l). Can t ila 3(x) 2 **/?

    is rod Canr t 2t

    or uaon , uset do a kld F and polroials tn Flx

  • Tm F) is prndi pl idorl doan PTb), a k T P Ta an la Te PC*] T=(o),* is prnpal T=(o)

    T (o), clasvt a poly n m )eL 0u ma re.

    m() CL SSu Pr t P6e) eT\(la)

    d senna m (e)

    a ma in lL

    = q (z)m (x)+ (x) Ja r(x)é da m(z) 9 (x)

    (a)eC si e >f(x)-q(* )m (oe )

    m (x). (art In)

    Colla In F(3) has bim (n() lo), a

    = *ta,, x^** amone daA.

    m( m (x) ioaals (m(T)), (z) au dshine

  • Divisb seR S dvdss s cls 3'e

    seCr) -Priniga ad attd .

    clo re, nLe $Lu)

    u H clu Ofr (

    D

    ta,g ispoyoniol Ac). al, alg C) el, clg - cl -(,3) 3) Smo

    San at , d Ia dve Pra

    8,8 cds

    A'8'1&,Fl)is&n d,d' drktt a a un See saa ho

    F-3 P

    exists

    P

    Ju s penpal (F6) is a PId) , T- ( )

    (R),lass -()- &F()o V Can dao

    cl,ey > cc',ecc" X=alrba acc. bc c'= clactbc')

  • maEuu) FheAeS, p(x)e F) oa pret -7

    no ainds of ma n dscre

    T) I2). .. q (a)

    n22

    Sln P) () - I~

    1 a (ne P«}+ G(x) q(«) S a.

    al) fc)k (a b(x)g (e) k(x )

    ahr egk,la, k halht be = (a kt be)f

    en

    P ()

    To s8 d2moe geneca

    Su icuals fa, g ) = (?())+ (g(*o)

    Evec T,S 1Is In R I+Tiirjl iel,jeJa dls ToS

    tn R

    Pncpa Sum a{

    (),y) 6), gPM) (R-goe) (A() T

    aX ra. = a,\c+ a,-0, z* tQ,0

    /o.eF . c sca oc prena ds Ponaal s

  • 0 opude qc&8,3) , kd dauis To

    A&2s 3 , 3)

    is ph e man ln now

    (3 2 + ) )

    Exe (P,g) ad (,)ha

    u sa

    remasne

    (,3) - nea+t

    ( o neAnailn

  • ramp F=7/^= 5 -f (2e) 2e 2 x2z+, 1(«)z-**l.

    Rd (¢a),z(z?)) emann en

    (xt2A 2x tl, x'-x+1)

    -z +l, T* -+x 3=0 3+X*

    cdf(x)g () = *+l onc V

    -2x

    2) F F lo, 1) f(x)= *'t*°, (z)= 24X+|

    emaan (z, x'2)

    (z, z*)

    (z ) T

    &,au uladres pri

    2C

  • - 10

    Cplor vs M u mandn)

    F3 Ten Le sist udea polns omials 9r eFLx), m r o Su at o oan

    d O -

    +

    ola osef+ (f) 2ve

    has a w etndese

    Yemahd

    Pco + Wck -q e (4) 3e )

    CeQ (E) o set

    Such C Vwa

    -2e(t) , ht sdf +le)= +e)

    Coyess () : uqeseed O a «ll (o miols { Jkra n

    (-b2x+ b,x«b. 6,#0 E- 2

    6us F:>/e)

    4uaduic atx t+(f\

    A padns (a,a,)a¬F o,1

    telx a or all Rnea 6mh don

    &n 3 ic

    Cos F

    sis 2,z Cocs adad ,4) a;e F

  • -it n

    tus ? 4+4,r elAs s asre sìdus o ulo P

    a ossibu F/( )

    EKA F F=Q (= *+x*|. daaks as paromids o{ Ja at mogt

    at,

    o ul n K, mulkpas polno mus ,u sals a

    2

    rvi fon

    (24)( -3z) = 2-S -3x -2-Sx -3 (-r-)-2-5�x) = +S

    o 2 +*t(f)

    -3t 1f(x\)

    (24x)C-3)=4S 1n x+t). Xx = x-- x -l I R/r -

    -3-S*t2

    3*3* t 3 2) F F to,} o-2z+S

    =z+**l As.

    R: 7 Ix*xri) c(x)= xt*

    = -I = (wd 2)

    T s a kIS uMBr naAs

    IFCxl/x*zed

  • -12 Rmean yo ro c-a

    remain lA, a

    x -a) C

    eaF(x) F

    (x )+c (a-a) gla)+c =

    =0gla) +C =C &la) =C

    lx)- (z-a)g (*) * Pla) isla)

    (z) is avi des (x-a) ,0 remaln

    is la)

    e F Consfe pono is FLax3-e oses ane

    he)t(x -a) k(a) jkn, n, (spo sweh

    t(aa

    R/z-) S sa IR 2 2

    3

  • Com Son -13

    FLx a Bo au PTDs

    Invti Fl-)-F

    ae

    Sa Conste

    (n) (-a) posi unn

    ie Pornmial

    n m k a)=q(z )h(x)

    P° e ite u cGe polynd

    23,,7,1,1) P) at.

    eFa no e duiGle lnamdy

    (uille's la aeks ) (i&L elan.s)

    WM C

    z1- p.(x).. P«() PiPa i radobn

    o C Ju po s

    =(-) p P

    T =a a P P(z)

    T moade esu,

    ced ed,q) ecm ,) ec ,3)

    opi , an+bm Som a, So a(« ),

    alx){r\r b(x)g)