Документ взят из кэша поисковой машины. Адрес оригинального документа : http://www.mccme.ru/pdc/2007/xtra/Sadykov/sadykov-research%20statement-Russian.pdf Дата изменения: Wed Oct 24 17:36:30 2007 Дата индексирования: Thu Jan 15 21:07:24 2009 Кодировка: Поисковые слова: m 8

, . "" .. , n (z1 , . . . , zn ) Cn , Qi ,..., z1 z f (z1 , . . . , zn ) = 0,
n

i = 1, . . . , n.

(1)

Q1 , . . . , Qn ­ Cn . (1) -- (. [3], 6). xi Pi ()y (x) = Qi ()y (x), i = 1, ..., n. (2)

Pi , Qi ­ n , = (1 , ..., n ), i = xi i . x . -, -, . , (1) . , xi = ezi , i = 1, . . . , n y (x1 , . . . , xn ) = f (log x1 , . . . , log xn ), (1) Qi (x1 , . . . , xn )y (x) = 0, i = 1, . . . , n, xi = xi i . , x (2) P1 . . . Pn 0. , (2) C[x1 , . . . , xn ] Dn . , . , .

1


. ( 21- ) . . . , , --, , -- . . Dn x1 , . . . , xn , x1 , . . . , xn , Dm ­ y1 , . . . , ym , y1 , . . . , ym . xj = xj xj 1 j n, yi = yi yi , 1 i m. x = (x1 , . . . , xn ) y = (y1 , . . . , ym ). A = (aij ) Z(n-m)вn n - m, (1, . . . , 1), B Znвm = (bj i ) m, , A · B = 0. 1 j m bj = (bj 1 , . . . , bj m ) Zm j - B . g () B . i = 1, . . . , m c = (c1 , . . . , cn ) Cn
|bj i |-1

Pi =
bj i <0 l=0 bj i -1

(bj · y + cj - l),

(3)

Qi =
bj i >0 l=0

(bj · y + cj - l),

(4) (5)

Hi = Qi - yi Pi ,

bj · y = m bj k yk . Hi , k=1 LB = {B · z : z Zm } c. di = bij >0 bij = - bij <0 bij Hi . 1. Dm : Horn (B , c) = H1 , . . . , Hm Dm . b(i) B . u Rn u = u+ - u- , (u+ )i = max(ui , 0) (u- )i = - min(ui , 0). i = 1, . . . , m Ti = x+ - x- , 2
b
(i)

b

(i)


vn v1 v x = x1 · · · xn . u LB u u Tu = x + - x - .

, LB . 2. , LB , IB = Tu : u L
B

C[x1 , . . . , xn ].

, , A, IA = Tu : u ker Z (A) C[x1 , . . . , xn ]. I = T1 , . . . , T
m

C[x1 , . . . , xn ].

I . , m = 2 I . , , m > 2. IA . A- A · c :
n

HA (A · c) = IA +
j =1

aij xj xj - (A · c)i : i = 1, . . . , n - m Dn .

n A · - A · c j =1 aij xj xj - (A · c)i : i = 1, . . . , n - m . A- , [8] , (., , [9]). , (. [15]). . ; , , . [4] -. , [10, 11]. [12] [16]. A- [17]. , , .

3


: 3. . 4. . , , , , . 5. . , . . 6. . , / , . . , --. . , (. [7],[2] and [14]). 7. , , . . 8. . 9. m = 2 ij = min(|bi1 bj 2 |, |bj 1 bi2 |), 0 bi , bj Z2 , ,

1 i, j n. ij , bi bj . 4


10. B ­ n в 2 , , b1 , . . . , bn b1 +· · ·+bn = 0. c Cn Horn (B , c) HB (c) . , rank (HB (c)) = rank (Horn (B , c)) = d1 d2 - bi , b bi , bj Z2
j
. .

ij = g · vol (A) + bi , b
j
. .

ij ,

B , , .

11. . . - : S P ­ m + 1 , E ­ Fm 1 (P \ S ) Aut(E ) ­ = E . Fm ­ m . , Fm E , a, b E Fm , a b. E P \ S , E S , ? - P 0, 1, ( ). [13]. [0, 1] . , . v , e , f , v - e + f . . - . ( .) . S, , , S, . , E , E . S S . , S . E S S . , , , . 5


12. T ­ . : (i) T . (ii) - T . (iii) - T . (iv) T ­ , 2-, . . (, ), () . , , . , , , , , , .


[1] . .. . . 412, (2007), . 4, 1 - 3. [2] . .. . . 198, (2007), . 9, 59 - 80. [3] .. . . .: , 1967. [4] F. Beukers and G. Heckman. Monodromy for the hypergeometric function n F (1989), 325-354.
n-1

, Invent. Math. 95

[5] E. Cattani, A. Dickenstein and B. Sturmfels, Rational hypergeometric functions, Compositio Math. 128 (2001), 217-240. [6] P. Deligne and G.D. Mostow. Monodromy of hypergeometric functions and non-lattice integral monodromy. Publications Mathematiques de l'IHS 63 (1986), 5-89. [7] A. Dickenstein, L. Matusevich and T. Sadykov. Bivariate hypergeometric D-modules. Adv. in Math., 196, no. 1 (2005), 78-123. [8] I. M. Gelfand, M. I. Graev, and A. V. Zelevinsky. Holonomic systems of equations and series of hypergeometric type. Dokl. Akad. Nauk SSSR, 295(1):14­19, 1987. [9] I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky. Hypergeometric functions and toric varieties. Funktsional. Anal. i Prilozhen., 23(2):12­26, 1989. [10] M. Kato. Appel l's F4 with finite irreducible monodromy group, Kyushu Journal of Mathematics 51:1 (1997).

6


[11] M. Kato. Appel l's Hypergeometric Systems F2 with Finite Irreducible Monodromy Groups, Kyushu Journal of Mathematics 54:2 (2000). [12] M. Kato and M. Noumi. Monodromy groups of hypergeometric functions satisfying algebraic equations, Tohoku Math. J. 55 (2003), 189-205. [13] S. K. Lando and A. K. Zvonkin. Graphs on surfaces and their applications. With an appendix by Don B. Zagier. Encyclopaedia of Mathematical Sciences 141. Low-Dimensional Topology II. Springer-Verlag, 2004. [14] F. Larusson and T. Sadykov. arXiv.math.CV/0607773 (2006), 11 pp. Dessins d'enfants and differential equations.

[15] Mutsumi Saito, Bernd Sturmfels, and Nobuki Takayama. GrЁ obner Deformations of Hypergeometric Differential Equations. Springer-Verlag, Berlin, 2000. [16] S. Tanabe. Logarithmic vector fields and multiplication table, math.AG/0602301. [17] U. Walther. Duality arXiv:math.AG/0508622. and monodromy reducibility of A-hypergeometric systems,

7