Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://higeom.math.msu.su/course_papers/aiz5.pdf Äàòà èçìåíåíèÿ: Sun Mar 18 01:07:08 2012 Äàòà èíäåêñèðîâàíèÿ: Sat Apr 9 23:52:02 2016 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: m 13

.. -

502 "F- " - -. , ..



1 2 3 f - h- 4 - 5 f - h- 6 .. . 7 . 8 K- - . 9 10 Ul 11 Ul 12 13 2 2 5 7 9 12 12 13 14 17 19 20 21

1


1



. 2 3. , .., , . - , [8]. [6] , . .. . f - h- (n - 1)- , n . f - -. f - . - [4]. ( 4 ­ 7) . .. A ( ) ord A, K .. pos K . ord pos K = cone K . .., . f - , , f - . [7], . .. . , ... 3 , K -. ( 8-12) K - . f - K , K -, f - f - . s . .. [3]. s(K ) K Ul , Ul -- , [5]. , s , , Ul l . s. Ul . s . , . .

2



1. M -- . M K , K 2M , : 1) K , K . 2) K . K . M , K -- . M V (K ). , , . dim | | - 1. K , , dim K . 2


K . pos K . 2. K , . 3. K -- . K , K . K {1 , . . . , l } K , . 4. K1 , K2 -- . K1 K2 V (K1 ) V (K2 ). = 1 2 , 1 K1 , 2 K2 . 5. K -- K . K , K , = , K . link( ), , . linkK ( ). . = 1 2 K . linkK = linklink , .
K

1

2 , ..

6. (n - 1)- K , (link( )) = 1 - (-1)n-|| K . (n - 1)- , (K ) = 1 - (-1)n . . . , . K. K = Z[K ]. K1 · K2 = K1 K2 K. link(v ), link(v ) = link({v }), d. dK = . , . 1. d(K1 · K2 ) = dK1 · K
2 v K

+ K1 · dK2 .

. v , . v K1 K2 v K1 v K2 . d(K1 K2 ) = link(v ) = linkK1 K2 (v ) + linkK1 K2 (v ) = =
v K
1

v K1 K

2

linkK1 (v ) K2 +


v K
2

v K1

v K2

K1 linkK2 (v ) = d(K1 ) · K2 + K1 · dK2 .

, v K1 K1 K2 link(v ) = { K1 K2 , v , / K1 , K2 } = linkK1 (v ) K2 . K fi = #{ K, | | = i} -- i - 1. max k = dim K - 1 = n. f F K Z[t] Z[, t] FK (, t) = f0 n + f1 n-1 t + . . . + fn tn , fK (t) = FK (1, t) = f0 + f1 t + . . . + fn tn .
fk =0

3


2. f (K, d) (Z[t], t ). F , (K, d), (Z[, t], t ).

. f .
K1 K2

fK1

K

2

=

t

| |

=


1 K1 ,2 K
2

( t
|1 |+|2 |

=



)( t
|1 |

·



) t
|2 |

= fK1 · fK2 .

1 K1

2 K2

n1 = dim K1 + 1, n2 = dim K2 + 1. , , dim(K1 · K2 ) + 1 = n1 + n2 = n. (t) , FK (, t) = fK dimK +1 F (, t) = fK () t ·
n1 +n2

K1 K2

1

K2

[ () ][ () ] t t = fK1 · n1 · fK2 · n2 = F

K1

·F

K

2

.

, . F . , K - , K . ( F
dK

=F =



) link(v ) =


v K

Flink(

v)

=





t|| t

dim(link(v ))+1-| |

= t
| |-1 n-| |

v K

v K link(v )

t

| | n-1-| |

=

| | n-1-| |



= =


K,v



= FK . t

v K, K,v ,v K /

K,v , = \v



| |t

| |-1 n-| |



=

K

. = 1, f . , dl K =
| |=l

l! · link( ), .

1. K = n-1 , .. , n- . n-1 = 1 . . . 1 , fn-1 (t) = (1 + t)n ,
n

, k - n- (n) n k . , Fn-1 (, t) = ( + t) . v n link v = n-2 , dn-1 = nn-2 . 2. n (n + 1)- [n + 1], , [n + 1]. d( n ) = (n + 1 1) n-1 , f - F n (, t) = (( + t)n+1 - tn+1 ). H- HK (, t) = FK ( - t, t). t = t + . HK = HdK . 3 ( -). K - , dim K = n - 1, HK (, t) = HK (t, ). . n. n = 1 . , K , . FK = ( + 2t), HK (, t) = + t . . , L dim L n - 2 . K n, . link(v ) 4


, , , Hlink(v) (, t) = Hlink(v) (t, ). v K , HdK (, t) = HdK (t, ), HK (, t) = HK (t, ), ( ). HK (, t) = HK (t, ) + P ( - t) = HK (t, ) + C · ( - t)n . , H - . C . = -1, t = 0 H (-1, 0) = H (0, -1) + C (-1)n , F (-1, 0) = F (1, -1) + C (-1)n , (-1)n = 1 - (K ) + C (-1)n , C = (-1)n ((K ) - (S n-1 ). , C = 0 . 1. K -- (n - 1)- . HK (, t) = HK (t, ) + ((K ) - (S
n-1

) · (t - )n .

. (n - 1)- k (n - k - 1)-. K . , h- . , , HdK (, t) = HdK (t, ), HK (, t) = HK (t, ), HK (, t) = HK (t, ) + P ( - t) = HK (t, ) + C · ( - t)n , C = (-1)n ((K ) - (S n-1 ). . , h-.

3

f - h-

7. [4], K n - 1, n . .. c : V (K ) [n] = {1, . . . , n}, . . . , . . , , . . . 4. K K. . , , , . K -- n 1, . . . , n. , K T [n] = {1, . . . , n}, c( ) = T . 8. ( ) f - K . T [n]. fT = #{ K, T }.

5


f - fl (K ) = fT .
T [n],|T |=l

9. ( ) h-. T [n]. hT = (-1)|T |-|R| fR .
RT

- , fT = hR .
R T

5 ( -). K -- n - 1. h- hT = h
[n]\T

.

. c K n . T fT t [n]\T . 2n f (1 , 2 , . . . , n , t1 , t2 , . . . , tn ) = f (, t) =
T [n]

t = t1 t2 . . . tn , i = 1 i T , i = 0 . n 12 T . h(1 , 2 , . . . , n , t1 , t2 , . . . , tn ) = h(, t) = hT t [n]\T . f - h- h(, t) = f (-t, t). , f (, . . . , , t, . . . , t) = F (, t), h(, . . . , , t, . . . , t) = H (, t). h- h(, t) i ti . , , h-. -- . n = 1. , h{0} = h{1} , , . . , n - 1 . K n - 1. fK (, t) = ti
v K,c(v )=i T [n]

T

flink(v) (1 , . . . , i , . . . , n , t1 , . . . , ti , . . . , tn ).

, , ( ) n + ... + fK (, t) = flink(v) (1 , . . . , i , . . . , n , t1 , . . . , ti , . . . , tn ). t1 tn i=1
v K,c(v )=i

- t, t t, h- ) ( + ... + + + ... + hK (, t) = hK (, t) = t1 tn 1 n =
n



h

link(v )

(1 , . . . , i , . . . , n , t1 , . . . , ti , . . . , tn ).

i=1 v K,c(v )=i

, hK (, t) = hK (t, ), t. h(, t) = h(t, ) + P (, t), P = 0. 6


, P P ( , t) =


T [n]

pT t

T

[n]\T

.

, , , P (, t) = n n = c (i - ti ). h(, t) = h(t, ) + c (i - ti ). c, 1 = . . . = n = -1, t1 = . . . = tn = 0. h(-1, 0) = h(0, -1) + c(-1)n , H (-1, 0) = H (0, -1) + c(-1)n , (-1)n = 1 - (K ) + c(-1)n , c = (-1)n ((K ) - (S n-1 )) = 0. .
i=1 i=1

2. M -- n - 1. h- hT = h
[n]\T

+ ((M ) - (S

n-1

))(-1)

n-|T |

.

. M -- , n . h(, t) = h(t, ) + c (i - ti ), c = (-1)n ((M ) - (S n-1 )),
i=1

. t .

T

[n]\T

, -

4

-

10. A B -- . A B .. A B , c d A B , () c A, d A, c A d () c B , d B , c B d () c A, d B . A B .. A â B , (a, b) (c, d) A B , () a A c () a = c b B d. A â B .. A â B , (a, b) (c, d) A â B , a A c b B d. . n n . n m n + m , n m nm . B 1 = 2. .. B n = B 1 â . . . â B 1 .
n

.. [n], . S .. , , . S .. ( ). A · B = A B S . , , . S ... . . .. A ord(A), A. , A. . 6. ord(A · B ) = ord(A) ord(B ). 7


7. d(ord A) =


v A
ord(A>v ) ord(A


),

A>v = {w A, w > v }, A

= {w A, w < v }.
-

. link(v ) = { ord(A), v } = {- + , / A>v } = ord(A>v ) ord(A
- A
+

- Av .

, S dA =


v A

, - d(A · B ) = dA · B + A · dB , A ord(A) (S , d) (K, d). , (S , d) (Z[t], t ), ord f - A fA = ford(A) . f -, tk , A k . . , ord n = n-1 , .. . fn = (1 + t)n . , K, ord(A B ) = ord(A) ord(B ). K1 , K2 -- . K = K1 K2 V (K1 )âV (K2 ). p1,2 V (K ) V (K1 ) V (K2 ). K K , p1 ( ) K1 v V (K1 ), p2 (p-1 (v ) ) K2 . 1 8. ord(A B ) = ord(A) ord(B ). . ord(A B ) -- = (a1 , b1 ) < (a2 , b2 ) < . . . < (am , bm ), a1 = . . . = an1 < an1 +1 = . . . = an1 +n2 < . . . < an1 +...+nk-1 +1 = . . . = an1 +...+nk , b1 < . . . bn1 , bn1 +1 < . . . < bn1 +n2 , ... bn1 +...+nk-1 +1 < . . . < bn1 +...+nk . p1 ( ) = {a1 , an1 +1 , . . . , an1 +...+nk-1 +1 } A. { , a p( ), / {bn1 +...+nl +1 , . . . bn1 +...+nl+1 }, a = an1 +...+nl +1 . -- B . A B ord(A) ord(B ). p2 (p-1 (a)) = 1 9. K1 K2 - . fK1 -- fK2 (t) - 1 fK1 . . fK1
K2 K2

(t) = fK1 (fK2 (t) - 1)

(t) =


K1 K
2

t

||

=
1 ,...,
| |


K1 ,1 ,...,
| |

t
K2 ,i =

|1 |+...+||

|

|

=
K2 , =

=


K
1



=

t

|1 |+...+||| |


K1

t
| |

| |

= = fK1 (fK2 (t) - 1).

K2 ,i =

=



(fK2 (t) - 1)

| |

K1

: .. fAB (t) = fA (fB (t) - 1). K = K1 K2 . |K1 | = m. L = K2 m -- m K2 . K1 L = . . . K2 . . . L, K2 , 8


- . , K1 K2 =


K1

L L.

K - . : . K1 K2 - , K1 K2 . K1 K2 K2 K1 , , .. .

5

f - h-

f - . .. -- . p1 , p2 -- . L = Q[t] L = Q . , L 1, t, t2 , t3 , . . .. f - .. L, fA = (f0 , f1 , . . . , fn , 0, 0, . . .). 10. w : L L L , .. A B fAâB = w(fA , fB ). . Bnm n â m. , Bnm , p1,2 : Bnm n, m . anm (t) Bnm . anm (t) = t|| . w,
p
1,2

Bnm ( )=n,m

w(v , u) =
AâB | |


i,j >0

vi uj aij (t) + v0 u0 .

(1)

fAâB =

t

=


1 A,2 B 1 â2 AâB

t

| |

=





t

| |

=

1 A, 2 B B|1 ||2 |

= , w .

fi (A)fj (B )aij (t) + f0 (A)f0 (B ) = w(fA , fB ).

i,j >0

11. w L . (L, w) . . w(w(v , u), y ) = w(v , w(u, y )) w(v , u) = w(u, v ) v , u, y L. L. f - .. . w(w(fA , fB ), fC ) = w(fAâB , fC ) = fAâB âC = w(fA , fB âC ) = w(fa , w(fB , fC )) w(fA , fB ) = fAâB = fB âA = w(fB , fA ). , (1). e = f1 = 1 + t. f - .. w(e, fA ) = f1âA = fA . . B L {bij }i,j 0 . i-1 (i) b00 = 1, b0j = bi0 = 0, bij = (-1)k k (i - k )j i, j 1. B , ..
k=0

9


1, t, . . . , tk , . : fK = B fK , K - , K -- (.[2]). C . , (1 + t). fcone K = C fK , K - , cone K -- , K . 3. v , u L C B (v · u) = w(C B v , C B u), v · u . , C B (L, ·) (L, w). . K -- . , pos(K ) .. . ord pos(K ) = cone K . . K L C B (fK ·fL ) = C B fK L = C f(K L) = fcone(K L) = ford pos(K L) = ford(pos K âpos L) = fpos(K )âpos(L) = w(fpos(K ) , fpos(L) ) = w(ford pos(K ) , ford pos(L) ) = w(fcone K , fcone L ) = w(C B fK , C B fL ). , f - , . , 1 C B e = 1 + t (L, w). . f - .. L. f - n-1 n . : w(1, 1) = 1, w(1, tQ(t)) = 0 Q(t). , â A = .. A f = 1. 3 f - ... p1 p2 t + 1. w(p1 (t), p2 (t)) = w(p1 (-1) + (t + 1)q1 (t), p2 (-1) + (t + 1)q2 (t)) = = w(p1 (-1), p2 (-1)) + w(p1 (-1), (t + 1)q2 (t)) + w((t + 1)q1 (t), p2 (-1)) + w((t + 1)q1 (t), (t + 1)q2 (t)) = = p1 (-1)p2 (-1) + p1 (-1)q2 (0) + q1 (0)p2 (-1) + w(C q1 , C q2 ) = = p1 (0)p2 (-1) + p1 (-1)p2 (0) - p1 (-1)p2 (-1) + w(C B B
-1

q1 , C B B

-1

q2 ) =
-1

= p1 (0)p2 (-1) + p1 (-1)p2 (0) - p1 (-1)p2 (-1) + C B ((B

q 1 ) · (B

-1

q2 )).

w . . wv = w(v , ·) E nd(L) v L, wA = wfA .. A. 1. wv
+y

= wv + wy ,

wAâB = wA wB = wB wA .

. . wA (wB (u)) = w(fA , w(fB , u)) = w(w(fA , fB ), u) = w(fAâB , u) = wAâB (u) u. . 2. wB 1 = t(1 + t)
d dt

+ id.

. fA .. A, L. wB 1 (fA ) = fB 1 âA . B 1 â A (0, a1 ) < . . . (0, al ) < (1, b1 ) < . . . < (1, bm ), a1 < . . . < al b1 < . . . < bm -- A, al b1 . B 1 â A M1 M2 . , 0 ( l = 0), -- . . b1 < . . . < bm , fM1 = fA .

10


fM2 (t) =


1,2 A,1 = max(1 ) min(2 ) v =max(1 ),1 =1 \v

t

|1 |+|2 |

=
v


A,1

t1+
2

|1 |+|2 |

=


v

=



t · fA v A

A>v

(t) =


v A

t · fAv (t) = d fA (t). dt

v A

t(1 + t) · fAv (t) = t(1 + t)fdA (t) = t(t + 1)

fB .

1

âA

d = fM1 (t) + fM2 (t) = fA (t) + t(t + 1) dt fA (t). -

t(t + 1)

d dt

.

4. , , B -- , C -- t + 1. wC B f (t) = f (). . w B C . vi (t) = (t + 1)i , i = 0, 1, . . .. C B vi (t) = C B fi-1 (t) = fC -1 (t) = fB i (t). wC B vi = wB i = wB 1 â...âB 1 = i i wB 1 = (id + )i = vi (). vi (t) , f L. . 12. w(C B v , u) = v ()u(t), w(C v , u) = [B
-1

v ]()u(t),

C B (v (t) · u(t)) = v ()[C B u](t). , f - . L -- Q[, t] . 13. w : L L L , .. A, B F
AâB

(, t) = w(FA (, t), FB (, t)).

. L L = Li . w P (n) Ln Q(m) Lm i=0 (t) w(P (n) (, t), Q(m) (, t)) = n+m-1 · w(P (n) (1, t), Q(m) (1, t)) . , , w L. . 3( L. B ) t Ln B P (, t) = n (B P (1, t)) . C -- ( + t). . 14. C B (L, ·) (L, w). . (L, w) (L, w) , deg P - 1. 11


6

.. .

11. r .. A , a, b A, a b {a = c1 < c2 < . . . < ck = b} k = r(b) - r(a) + 1. , .. 0 r(0) = 0. , , rk. .. , , . 1. K ... . 2. ord A .. , . ord A . f - .. f - . 3. .. , rk(a, b) = rk a + rk b. L f - {eS } S . f - .. ( ) -- , , f (A) = fS eS L (, fS = 0, S {0, 1, . . . , rk A}, , ). w : L L L w (f , g )S = fT · gR ,
T ,R- T +R=S S

T , R -- n = |S |, , S + T -- {i1 . . . in } + {j1 . . . jn } = {(i1 + j1 ) . . . (in + jn )}, T R -- , . 15. f (A â B ) = w (f (A), f (B )).

. S = {l1 , . . . , ln }. w (f (A), f (B ))S (i1 , j1 ) < . . . < (in , jn ) A â B , rk ik + rk jk = lk . {rk i1 , . . . , rk in } {rk j1 , . . . , rk jn }, .

7

.

K , L -- . K â L. = {(i1 , j1 ), . . . , (il , jl )} , {i1 , . . . , il } -- K , {j1 , . . . , jl } -- L, , ik im jk jm k , m = 1, . . . , l. 5. K , L fK F
âL

= w(fK , fL ), = w(FK , FL ).

K âL

12


. . , K L K â L Bkl . K â L , -- Bkl p1,2 ( ) .
A={(i1 ,j1 ),...,(il ,jl )} p1 (A)= K,p2 (A)= L |A|

t

=





t|

A|

=


i,j

fi (K )fj (L)aij (t)

=

w(fK , fL ).

K A â L

w . . K 0, 1, . . . , k - 1, L -- 0, 1, . . . , l - 1. : 0, 1 . . , . , â . , . . , : (v , w) c(v ) + c(w). , K â L -- . , . 16. K, L f (K â L) = w (f (K ), f (L)).

8

K- - .

12. K -- m , (X, A) -- . K (X, A) X m , (X, A) = {(x1 , . . . , xm ) X m , xi A, i }. K - / (X, A)K = (X, A) X m .
K

K - Am . (X, A) -- , K - dimX X m . f - CX (x) = ci xi , ci
i=0

i . 17. (X, A)K , (X, A) C(X
,A)K

(x) = FK (CA (x), CX (x) - CA (x)) · CA (x)

m-n

,

FK -- f - K , m = |K |, n = dimK + 1. . (X, A)K M . c = c1 â . . . â cm M , ci A i ci / X , A i . M

13


(CX (x) - CA (x))|| CA (x)m-|| . (X, A)K C(
X,A)K

(x) = =


K

(CX (x) - CA (x))|| CA (x)m-|| =
n-| |

(CX (x) - CA (x))|| CA (x)

CA (x)m-n = FK (CA (x), CX (x) - CA (x)) · CA (x)m-n .

K

K - K . (X, A) ((X, A)K , Am ). K . 18. K1 K2 = K1 K2 , K1 K2 -- , . . , K1 K2 V (K1 ) â V (K2 ) = {(ik , jk )}, 1 = p( ) -- K1 , i {jk } K2 , (i, jk ) . K1 K2 i i {i1 , . . . , ik } = 1 K1 21 , . . . , 2k . K2 m2 K1 K1 K2 ((X, A) , A ) (X, A) X m1 âm2 . x = (x11 , . . . , x1m1 , . . . , xm2 1 , . . . , xm2 m1 ) X m1 âm2 . , , x (X, A)K1 K2 K1 K2 , xij A, i i (i, j ) . 1 = {i1 , . . . , ik } K1 21 , . . . , 2k K2 / i xij A, i 1 i 1 j 2 . : 1 K1 / / i i i 1 2 K2 (xi1 , . . . , xim2 ) (X, A)2 , i 1 (xi1 , . . . , xim2 ) Am2 . : 1 K1 / , i 1 (xi1 , . . . , xim2 ) (X, A)K2 , i 1 (xi1 , . . . , xim2 ) Am2 . / , x ((X, A)K2 , Am2 )K1 . . K - . 13. - K ZK = (D2 , S 1 )K . , (D2 , S 1 ) C. , ZK Cm . ZK T m Cm . , m- . . K T T m . (x1 , . . . , xm ), xi = 0 i . . 14. s(K ) T m ZK . -- s(K ) .

9



15. Zl -- l. v1 , . . . , vk Zl , p : Zk Zl , p(ei ) = vi , Zl . : v1 , . . . , vk Zl .

14


K [m] Zl : [m] Zl , i vi , K {vi , i } . (K, ), -- K . l -- -- . , l K Zl . , l , . l = m : (i) = ei , ei . 19. r(K ) -- , . s(K ) = m - r(K ). . H T m , , T r . T m T m /H T m-r . H= = : (t1 , . . . , tm ) (t1 =
1, 1
1,1

...t

1,m m

, . . . , t1

m

-r,1

...t

m- m

r,m

),

. . .
m-s,1

... .. . ...
m

1,m . . .
m-s,m

: Z Zm-r . i ei , . . H , T . , |T : T T m-r . , . , , , . , Zm-r : T m T m-r , T T m , . K er , , , - . r Zm-r , . (K ) , K . . 20. m - (K ) s(K ) m - dim(K ) - 1. . s(K ) m - dim(K ) , , , . c : V (K ) {1, . . . , }. : V (K ) Z (v ) = ec(v) . , , s.

15


16. K L f : V (K ) V (L) , f ( ) L, K . , |f ( )| = | | K , f . . ChP, (K, ). (K1 , 1 ) (K2 , 2 ) f : K1 K2 , 1 = 2 f . , , . (l - 1)- Ul , l = 1, 2, . . .. Ul v Zl , . Ul : Ul Zl Zl , = id. l (Ul , ). . 19 : l K Ul m - s(K ). , K Ul . r(K ) = m - s(K ). , K L, K L . 21. K N (K ) (N ) s(K ) s(N ). , s . . . Li , Li Lj j > i , , K Li ( , Lj ). L- K i, K Li . , L- i-1 , L- . L- Ul L- (i) r(K ). , i + 1, L- . K N N Li , , , L(K ) L(N ). . L- . L1 L2 -- i i , i L1 L2 . i i , L1 L2 . , L- i i (i) L1 (K ) L2 (K ). : i Ui . , (K ) r(K ) dim K , 20. . 22. , . f - . . . , K N . f : K N g : N K . , ( g f (K ) ). . , f , .. . , f g , , 16


. . , K i, N , f , g . K N , . , g (N ). g (N ) N g (N ) K , , , N , , K . i, , N i. . . . 23. 1) f : K N K , f (linkK ) linkN f ( ). 2) f : K1 K2 , f id : K1 N K2 N . 3) K1 K2 , K1 N K2 N .

10

Ul

X (k) k - X , X (k-1) k . l-1 -- (k - 1)- l . , l -- l , (1) k . , l = Kl -- l . 24. U
(1) l (k)

K

2l -1

.

. K2l -1 Ul . S [l]. vS ei . l- eS , S 1, 0, eS =
i S

, .. (eS , eT ) Zl S = T . . S T , S T S1 . . . Sl [l]. , , eSi . S T , S T = , eS T . (eS , eT , eS T ) (eS , eS + eT - eS T , eS T ) = (eS , eS T , eS T ), , . S T = , . 27. (1) Ul K2l -1 . v Zl 2, v (mod 2). (0, . . . , 0) , 2, 2l - 1 . , . v1 v2 (mod2), ( (v1 + v2 )/2 vi ). K2l -1 . 25 (). r(Kp ) = log2 (p + 1). . Kp Ul , K2l -1 . l, , , 17


p

2l - 1. . p

-- , . , () , Kp . 6. -- . r() = log2 ( () + 1).

. K () , 21, r() r(K ). , Ul , K2l -1 , 2l - 1 . , () 2r() - 1. log2 ( () + 1) r() log2 ( () + 1), . 26 (). K . r (K ) log2 ( (K ) + 1).

. K (1) K . , 21, r(K (1) ) r(K ). K (1) -- r(K (1) ) = log2 ( (K (1) ) + 1) = log2 ( (K ) + 1). Ul (2). -- Zl . 2 , Z2 . 27. Ul
(2)

U

(2) l

(2).

. 2. Zl , Zl , , , 2 (3) (3) . Ul (2) Ul . Zl Zl . , , 2 Zl , Zl . 2 3 â l- 1,1 . . . 1,i1 . . . 1,i2 . . . 1,i3 . . . 1,l A = 2,1 . . . 2,i1 . . . 2,i2 . . . 2,i3 . . . 2,l 3,1 . . . 3,i1 . . . 3,i2 . . . 3,i3 . . . 3,l .. , Z2 , i1 , i2 , i3 . 3 â 3 0 1 , ±1. Z A , ±1. . 1. , ±1, , , , , . ( ) 100 023 , , (0 1 1). ±1. 2. 24, . . , 2 s . 18


11

Ul

Ul , [DJ],[vdK]. , . 28. Ul (l - 2)-. Ul linkUl (l - | | - 2)-. Ul -. Ul . 29. 1) Ul , | | = k . linkUl Ul-k . 2) Ul Uk Ul+k . . Ul , k . , , -- . 1) Ul-k linkUl . (v1 , . . . , vl-k ) Ul-k (0, . . . , 0, v1 , . . . , vl-k ) Ul . . , linkUl . f : linkUl Ul-k . D -- , . .. D , p : Zl Zl /D Zl-k . , p = linkUl Ul-k . -- , Zl . , Zl ( l - k ). p( ) = p( ) p( ) = p( ) {0} Zl-k , , , l - k , , Zl-k . . 2) f : Ul Uk Ul+k . v Ul Uk . (v , 0, . . . , 0), v Ul , k f (v ) = (0 . . . , 0, v ), v Uk .
l

, Ul Z Zl Zl Zk , Uk Zk -- Zk Zl Zk . , . , , , , . = (1 , 2 ) Uk Ul , 1 -- Zl , 2 -- Zk . f ( ) = f (1 ) f (2 ) -- Zl Zk . f .
l

30. K N r(K N ) r(K ) + r(N ).

. K Ul1 , N -- Ul2 , l1 = r(K ), l2 = r(N ). K N Ul1 Ul2 , , Ul1 +l2 . K N Ul1 +l2 , . 31. r(K L) r(K ) + dim L + 1. . l = r(K L). L. linkK L = linkL K linkUl , -- dim L + 1 Ul . linkUl Ul-dim L - 1, linkL K Ul-dim L-1 , , , K Ul-dim L-1 , .

19


12


r(K ) r(
(dim K ) (K )-1

32 ( ). ).
dim K (K )-1

. K

21.

cl -- Zl ( Zl ), 2 k k (, ). (k-1) . c s(p-1 ). 33. k 2 c r(
(k-1) p-1 l k -k+2

2l

+ k - 3,

)

log2 (p - k + 3) + k - 2.

. M -- Zl , |M | = c. 2 k - 2 v1 , . . . , vk-2 N . vi . NS . S [k -2] NS = N + i S vi , vi = w2 + , , w1 , w2 N w1 + i T iS vi = 0. ( ) w1 + w2 +
i S T

k M , -, . , 2l - 1 2k-2 , c - k + 2. 2l - 1 2k-2 (c - k + 2), c 2l-k+2 - 2-k+2 + k - 2, l-k+2 c2 + k - 3. , Zl . , 2 Zl , mod 2 Zl , , 2 . 34. cl = 2l 3 r(
(2) p-1

-1

,

) = log2 p + 1.

. 27, Zl , 2 . , Zl , 2 0 2 . 2l-1 . eA A . eA + eB + eC , eA + eB + eC = 0. , cl 2l-1 . 3 33 k = 3. , 25. 35. cl = l + 1. l . , l - e1 . . . , el Zl . , , . v , l - 1 . e1 , . . . , el-1 , v , v ±1. . l + 1 . , ei 2 v1 , v2 . 20


(±1, . . . , ±1), , , (v1 + v2 )/2 , , , v1 v2 , . 36.

r((p-1) ) = p, p r(p
(p-1) +1

) = p + 1.

. 35. . (p-1) (p) (p-1) (p) p+1 p+1 , r(p+1 ) r(p+1 ) = p + 1. r(
(p) p+2

) > p 35.

13



17. K -- . m, m- . m (K ) m- . . 1 -- . , m , ) i ( (m) (m-1) . 21. m Li = i -- -. Li m . ) i ( (0) i-1 , . , m = 1, , . . 37. k , n N K k (K ) kn (K ) , n (( )k (K ) ) (k-1) (k-1) r (K ) r |K | k (K )r(|K | ), k (K ) k (Ur
(K ) (m)

).

21



[1] . . . // . . .., .263, .1-26. [2] . . , . . . . , , 2004. [3] . . . . . , 2008. [4] M. Bayer, L. Billera. Generalized Dehn-Sommerville relations for polytopes, spheres and Eulerian partially ordered sets // Invent. Math. 1985. V.79, 1, P.143-157. [5] M. Davis, T. Januszkievicz. Convex polytopes, Coxeter orbifolds and torus actions // Duke Math. J. 1991. v.62., 2. P.417--451. [6] Alexander A. Gaifullin. Local formulae for combinatorial Pontrjagin classes // http://arxiv.org [7] G´ or ab Hetyei. Face enumeration http://www.math.uncc.edu/preprint/2004/ using generalized binomial coefficients //

[8] V. Klee. A combinatorial analogue of Poincar´ duality theorem // Canad. J. Math. 1964. V.16. e's P.517-531. [9] R. Stanley. Enumerative combinatorics, V.1. Wadsworth and Brooks/Cole, Monterey, California, 1986. [10] R. Stanley. Flag f -vectors and the cd-index // Math. Zeitschrift 216, 1994. P.483-499.

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