Hypergeometric systems II: GKZ systems

Transcription

1 Hypergeometric systems II: GKZ systems Uli Walther Aachen summer school, September 2007

2 Outline 1 Hypergeometric systems 2 Toric language 3 Solutions of A-hypergemetric systems

3 The hypergeometric differential equation 1 Our example: ( t 2 + c (a+b+1)t t(1 t) t ab t(1 t) )z = 0. Multiply with t 2 (1 t), write θ = t t, to get (standard) form (θ 1 + 1)(θ 1 + c)z = t (θ + a)(θ + b)z. General hypergeometric differential equation: v j 1 v j >0 l=0 (v j θ + c j l)z = t v j <0 v j 1 (v j θ + c j l)z l=0 Power series ansatz: z = k=0 a kt k shows a k v j 1 v j >0 l=0 (v j k +c j l) = a k 1 since θ(t k ) = kt k. So v j <0 a k /a k 1 Q(k). v j 1 (v j (k 1)+c j l) l=0

4 The multivariate case 2 Given v j 1 v j >0 l=0 (v j θ + c j l)z = t v j <0 v j 1 (v j θ + c j l)z (1) let v = (v 1,...,v n ), find A Z n 1,n with A v = 0, β := A c. vj>0 v j j v j <0 v j j l=0 φ = 0; n a 1,j x j j φ = β 1 φ; j=1 (2) n a n 1,j x j j φ = β n 1 φ. j=1

5 The multivariate case II 3 For our case, n = 4, v = (1,1, 1, 1) and c = (1,c,a,b) Pick A = , then β = (c 1, a, b), and so ( ) φ = 0, ( θ 1 + θ 2 ) φ = (c 1)φ, ( θ 1 + θ 3 ) φ = ( a)φ, ( θ 1 + θ 4 ) φ = ( b)φ. Elementary but painful check: [x β i N c i (x v ) i Sol(System 2)] [ i N c i t i Sol(Equation 1)].

6 The multivariate case III: GKZ systems 4 A Z d n, β Z d. assume NA pointed, and ZA = Z d ; O A = C[x 1,..., x n ], D A = WeylAlgebra(O A ) = O A, R A = C. toric ideal: Euler operators: GKZ system H A (β): I A = u+ u A u = 0,u Z d R A E i = n a i,j x j j. j=1 P φ = 0 P I A ; (E i β i ) φ = 0 i = 1,...,n. If d + 1 = n, Sol(H A (β)) Sol((Equation 1)).

7 Toric variety/polyhedral structure 5 A = ((a i,j )) = (a 1,...,a n ) Z d n integer matrix. π induces R A A C[t ±1 1,...,t±1 d ] with j t a j. ker(π A ) = I A = u + u A u = 0,u Z d R A S A = C[NA] = C[t a 1,,t a n ] C[t 1 ±1,...,t±1 d ]. π A gives presentation, S A = R/I A. V A = Var(I A ) C n R + A cvx plhdrl rtl cone, with faces {τ} τ R A (I A, { j } j / τ ), Z d -graded primes I A.

8 Examples 6 Ex. I: A = ( ) 1 2 0, R A = C[ 1, 2, 3 ] I A = , A I A, {1} ( 2, 3 ),{3} ( 1, 2 ), ( ). ( ) Ex. II: A =, R A = C[ 1,..., 4 ], I A = , , , A I A, {1} ( 2, 3, 4 ),{4} ( 1, 2, 3 ), ( ).

9 Torus action 7 Let T = (C ) d ; make map C n T C n via (ξ 1,...,ξ n ) (s 1,...,s d ) (s a 1 ξ 1,...,s an ξ n ), (1 τ ) j = 1 if j τ and = 0 else. Orb(1 τ ) := image of 1 τ T, a torus of dimension dim(τ). V A = Orb(1 A ) Zar = Orb(1 A ) cplx = Orb(1 τ ). τ face of R +A Invariant tangent vectors on T: {t i t i }. Under T Orb(1 A ), t i t i n j=1 a i,jx j j =: E i (mostly). Define: H A (β) = D A (I A, {E i β i } d i=1 ) M A (β) = D A /H A (β). Holonomic dim C (Sol(H A (β))) <.

10 Solving polynomials 8 Example: A = ( ) and β = (0, 1) give H A (β) = D A ( , x 1 1 +x 2 2 +x 3 3, 2x 1 1 +x ). Then the solutions to x 1 T 2 + x 2 T 1 + x 3 T 0 = 0, ρ 1,2 = x 2 ± x2 2 4x 1x 3 2x 1 span Sol(H A (β)). (True more generally.)

11 Singular locus of H A (β) 9 Rank: rk(m) := dim(sol(m)) in generic pt = dim C(x) (M C(x)). A-discriminant (for A with top row (1,..., 1)): consider x 2,1 T a 2, x 2,n T a 2,n = 0, x d,1 T a d, x d,n T a d,n = 0 Thm: A ({x i,j }) with A (x) = 0 iff system has unusual number of torus solutions. rk = const away from A (x) = 0. ( ) If A = then A (x) = x 1 x 3 (4x 1 x 3 x2 2)..

12 Nice hypergeometric series, I 10 Basic idea: assume: I A homogeneous, β = A v, v j { 1, 2,...}, Λ = ker Z (A). consider φ v = u Λ [v] u [v + u] u+ x v+u C[[x 1,...,x n ]] where [v] u = j (v j (v j 1) (v j u j + 1)). easy check: killed by H A (β). problem: not always a function.

13 Nice hypergeometric series, II 11 A homog., β generic, L a generic weight on R A, θ A = {x j j } n j=1. J L A = (D A gr L (I A )) C[θ A ] + (A θ A β), a radical ideal. Var(J L A ) = {v1,...,v vol(a) }. Theorem The following are a basis for Sol(H A (β)): { φ v = } [v] u x v+u JA L [v + u] (v) = 0 u+ u Λ Remark L gives regular triangulation T of A; to each simplex I belong vol(i) many of the φ v. In general: rk(h A (β)) vol(a). (homog: wiggle β, and then specialize. else: ideas from next part.) sometimes, J L A grl (H A (β)) C[θ A ] (sometimes) 2, more than vol(a) solutions: coming soon.