Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://higeom.math.msu.su/course_papers/bur4.pdf Äàòà èçìåíåíèÿ: Mon Apr 9 22:55:40 2012 Äàòà èíäåêñèðîâàíèÿ: Sun Apr 10 00:07:29 2016 Êîäèðîâêà:

, .
. .


. . , .

C ­ , f (x, y ). C , CP2 A . m n ­ x y . , m > n, C y A. [4], n m, . [4] C2 , . C
515.162 . . - .


1


Ej0 (E1 )
s

s

s

E
s s s

i

1

s

s

Ei
s s s

2

s

s

Eih s s s

sEj1

(A)

sEj

2

sEj

h

. 1: . , C A . , . . , , . [4] , A , . 1. D ­ A. , A 0, 1 T = ih . , Ei , ai . Ejk . g ­ , Ek D. k (g ) ­ g Ek . Eu1 , Eu2 , . . . , Eur ­ D. ui C[x, y ] : J (v1 , . . . , vr ) = {g C[x, y ] : ui (g ) vi }. 1 g Ek D k (g ) 0, g Ek , , g const Ek . . . Eu1 , . . . , Eur . , v = (v1 , . . . , vr ) Zr dimJ (v ) < . d(v ) := dim(J (v )/J (v -1)) 2


L(t) :=
v Z
r

d(v )t , 1 = (1, . . . , 1) Zr , t = tv1 · · · tvr . 1 1 r

v

v

, L(t) ­ . [1] {J (v )}
r

L(t) P (t) =

(1 - ti )
i=1

1-t

1

.

[1] , P (t) =
PC[x,y ]

t

(g )

d ,

= t1 u1 t2 u2 · · · tr ur , PV t V . (g ) t d
PC[x,y ] (g ) v

(g )



(g )

(g )



(g )

t
PC[x,y ]

d :=
v Zr 0

({g PC[x, y ] | (g ) = v })t

, {g PC[x, y ] | (g ) = v } ( [1]). = (Ei Ej ) ­ D, M = -1 = (mij ). E k Ek D. M . Ek gk ,
· ·

{gk = 0} Ek D. l mkl = l (gk ). [5] , Ejk Eil M . mk = (mku1 , . . . , mkur ). , ajk , , k - 3


. El al , h- , aih-1 aih . h = 1, . 1 , . , h 2, P (t1 , . . . , tr ) =
0kT

(1 - t

mk -(E k )

·

)

.

1 [2] , . 1, 1 . k = jk (f ), 0 k h , dl = g cd(0 , . . . , l-1 ), 1 l h + 1 , qs = ds /ds+1 , 1 s h . [4] dh+1 = 1, qk k Z0 0 + Z0 1 + · · · + Z0 k-1 . S ­ C ([4]). [6] [4] , 0 , . . . , h S , S = a0 0 + a1 1 + · · · + ah
h

0 a0 , 0 ai < qi 1 i h. [4] , dk | n. n nk = dk . gk , 1 y , degy (gk ) = nk , k , degy (k ) < n - nk ,
d f = gk k + k .

, gk k . g0 = x. , gh+1 = f . g0 , . . . , gh 4


f . Ck = {gk = 0}. [4] , 2 k h k = 0 Ck · E jk . C1 E l 1· , E j1 . . [5] g0 = x,
m n

·

g1 = y +
j =0

cj x j , c j C ,
i-1 -1

g

i+1

q = gi i + a

0 1 ···

i-1

g0 0 g1 1 · · · gi

+
(0 ,1 ,...,i )i ···
i

c

0 1 ···

i

g0 0 g

1

1

· · · gi i ,

a

0 1 ···

i-1

C , c

0 1

C (1 i h),


i- 1

j j = qi i , 0 j < qj (0 < j < i),
j =0

i = {(0 , 1 , . . . , i ) Zi

+1

| 0 j < qj (0 < j < i),
i

0 i < qi - 1,
j =0

j j < qi i }.

c0 1 ···i C , g0 , . . . , gh f . , 0. g1 = y · C1 E j1 , Ck k 2 . g
a0 a1 ...a
h+1

aa h+1 = g0 0 g1 1 · · · gh+1

a

5


0 a0 , 0 ah+1 , 0 ai < qi 1 i h. C[x, y ] ([6]). 2 k h + 1 Ck C , Eik-1 , Ck Eik-1 ([4]). g0 , . . . , gk-1 gk ([4]). d0 , d1 , . . . , k-1 . dk k k 1 k h + 1 C[x, y ], ga0 a1 ...ak , 0 a0 , 0 ak , 0 ai < qi 1 i k - 1. k - . a = (a0 , a1 , . . . , ak ), , k - . Ik ­ k - . 2 g = aIh a ga . El , l (g ) l (ga ) , g . . Gl ­ , g , l . , Gl El . , l = ik 1 k h - 1. , Eik . , ga , gb Gl Ca Cb Eik . . [4] (Eik Cp ) = dk+1 =q dp
k+1 qk+2

···q

p-1

,

k + 1 p h + 1. (Eik Ca ) = a i =
k+1

+a

k+2 qk+1

+a

k+3 qk+1 qk+2

+ · · · + ah q

k+1

···q

h-1

.

i dk+1

0 i k . [4] (1)

ik (ga ) = a0 0 + a1 1 + · · · + ak k + qk k (Eik Ca ) 6


ai bi , ak+1 , ak+2 , . . . , ah , bk+1 , bk+2 , . . . , bh , , ai = bi i k + 1. ik (ga ) = ik (gb ), a0 0 + a1 1 + · · · + ak k = b0 0 + b1 1 + · · · + bk k , , a = b, , ga gb . , h 2 al - aih-1 , aih . al al aih-1 , . , ga , gb Gl l (ga ) = l (gb ). , . F = ga . F gb El El . , Ca Cb h- Ejh , F Eih-1 Ejh Ejh . , ah = bh . i i = dh 0 i h - 1. , F Eih-1 . , ih-1 (ga ) = ih-1 (gb ), 1, ah = bh , , a0 0 + a1 1 + · · · + a
h-1 h-1

= b 0 0 + b 1 1 + · · · + b

h-1 h-1

, , a = b, . h = 1 . , ga , gb El , D . , a0 = b0 , a1 = b1 . 1. h. h = 1 2. , , l (g0 ) 0 l (g1 ) 0 l = j0 , j1 . , g0 = x g1 = y . C . Ej0 , j0 (x) = 0, j1 (x) = 1, , x = 0, 7


El l (x) > 0. , j1 (y ) = 1 y = 0, l l (y ) > 0. h 2. , Eih-1 , gh , Eih-1 . , Eih-1 . 2 gk , 0 k h. ih-1 (gk ) > 0. , Eih-1 , , Ck k h - 1, l (gk ) > 0 k < h l > ih-1 . ih-1 (gh ) > 0 , Ch . gh . Ejh , Ch . jh (gh ) < 0, , Ejh , gh , Eih , ih (gh ) = mih jh = jh (f ) = h > 0, , jh (gh ) 0. Ejh Ch . gh , . 1. . . N Ih ­ h. C[x, y ]N ­ , h ga , a N . t d =
PC[x,y ] =N Ih PC[x,y ] N <
N



t d.



2 t PC[x, y ]N . N 2

8


(PC[x, y ]N ) = ((C ) P (t1 , . . . , tr ) =
aIh

N -1

) = 0, C = C\{0}

t

(ga )

=
(g
ai i

=
0a0

t

(g0 0 ) 1ih-1 0ai
a

t 1 1-t
(g0 )

) 0ah

t 1-t

(g

ah h

)

= 1 1-t
(gh )

qi (gi ) (gi )

=

1ih-1

1-t

.

, (gk ) = mjk , 1 k h - 1 k - , [4] , qk (gk ) = mik 1 k h - 1, . h = 2. , PC[x, y ] . q q g2 = g11 + ag0 0 . f = g11 + g0 0 . V PC[x, y ] , t d =
PC[x,y ] V

t d



, V , t . W PC[x, y ] ­ {g0 0 g1 1 fi = 0}, 1 < q1 . ,
i

W . g . 2- : g=
i,k j
{g = 0}, = (0 , 1 , 2 , . . . , C
q1 -1

q1 -1

), s

g =
j =0

j j g1 i,k

ik aij k g0 g2 .

9


, t . 2 1- 2- . . 1-
q1 -1



g =
j =0

j j g1 i,k

q i aij k g0 (g11 + ag0 0 )k .

q j i g1 i,k aij k g0 (g11 + ag0 0 )k j , . . , ,

{g

j 1 i,k

ik aik g0 g2 = 0}, j < q1 ,

. :
j g1 i,k j ik aik g0 g2 = g1 s i+ 0 k = s ik aik g0 g2 .

{g = 0}, = (0 , 1 , 2 , . . .), s C g = g
j 1 s



s i+0 k=s

ik aik g0 g2 .

, t . g . , q i i+0 k=s aik g0 (g11 + ag0 0 )k s 1- . 2 t . , ,
j {g1 i+0 k=s ik aik g0 g2 = 0}, j < q1 ,

10


. , {g0 0 g1 1 (g2 + i g0 0 ) = 0}, i
i

C, 1 < q1 , W . [4] [5] , {f = 0} = 0, a 1
·

Ei1 . Y S k Y k - . · X = p0 p1 p2 E i1 , ps ­ Ejs s . g W D, .
2

W =
k 0

SkX =
s=0 k0

Skp

s

â
k 0

S k E i1 .

·


2

t d =
W s=0 k0



(S k ps )t = (1 - t
m
i1

km

js

(S k E i1 )t
k 0

·

km

i1

=

)-

(E i1 ) s=0

·

2

(1 - t
k=0

m

js

)-

(ps )

,
-(Y )



(S k Y )tk = (1 - t)

. ·

, (ps ) = (E js ).


[1] Campillo A., Delgado F., Gusein-Zade S.M.: The Alexander polynomial of a plane curve singularity and integrals with respect to the Euler characteristic, Int. Journal of Mathematics, 14 (2003), No. 1, 47­54. [2] A. Campillo, F. Delgado, S. M. Gusein-Zade: Multi-index filtrations corresponding to curves with one place at infinity and their Poincare series, arXiv:math/0506549.

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[3] Viro, O. Ya. Some integral calculus based on Euler characteristic. Topology and geometry Rohlin Seminar, 127­138, Lecture Notes in Math., 1346, Springer, Berlin, 1988. [4] M. Suzuki: Affine plane curves with one place at infinity, Annales Inst. Fourier, 49 (1999), 375­404. [5] M. Fujimoto, M. Suzuki: Construction of affine plane curves with one place at infinity, Osaka J. Math., 39 (2002), 1005­1027. [6] H. Pinkham: Courbes planes ayant une seule place a l'infini, Seminaire sur les singularites des surfaces, Ecole polytechnique, Annee 1977­1978.

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