aa r X i v : . [ m a t h . C A ] J a n Modified A -hypergeometric Systems Nobuki Takayama, Department of Mathematics, Kobe UniversityJune 30, 2007, January 20, 2008 revised
Abstract
We will introduce a modified system of A -hypergeometric system(GKZ system) by applying a change of variables for Gr¨obner deformations andstudy its Gr¨obner basis and the indicial polynomial along the exceptional hy-persurface. Since the work of Gel’fand, Zelevinsky, and Kapranov [3], studies of A -hypergeometricsystem (GKZ system) have attracted a lot of mathematicians, who want to un-derstand hypergeometric differential equations in a general way. We refer to thebook [9] on the status of the art in 2000, and the recent papers [4] and [10] andtheir reference trees on recent advances. We also note that these studies havehad fruitful interactions with frontiers of computational commutative algebraand computational D -modules.In this short paper, we will introduce a modified version of this A -hypergeometricsystem and provide a first step to study it. The original system is defined on the y = ( y , . . . , y n ) space and the modified system is defined on the ( t, x , . . . , x n )space with one more variable t . Let us sketch our idea to introduce the mod-ified system. We consider the direct sum of the A -hypergeometric system onthe y space and the D -module D/D · s∂ s on the s -space. For a weight vector w ∈ Z n , the original system restricted on the complex torus is transformed intothe modified system on ( t, x ) space by the map C n × C ∗ ∋ ( y , . . . , y n , s ) ( t w x , . . . , t w n x n , t ) ∈ C n × C ∗ (see [8] and [9] on this transformation). The transformed system can be nat-urally extended on C n +1 . Intuitively speaking, the variety t = 0 is analogousto the exceptional hypersurface of a blowing-up operation. We will study theindicial polynomial along t = 0 as a first step to make a local and global anal-ysis of the modified system. As a byproduct of our discussion on the modifiedsystem, we will also give a proof to the claim rank( H A ( β )) ≥ vol( A ) for non-homogeneous A . 1 Definition and Holonomic Rank of Modified A -hypergeometric systems Let A = ( a ij ) ij be a d × n -matrix whose elements are integers and w =( w , . . . , w n ) a vector of integers. We suppose that the set of the column vectorsof A spans Z d . Define ˜ A = a · · · a n · · · a dn · · · a dn w · · · w n . Definition 1
We call the following system of differential equations H A,w ( β ) a modified A -hypergeometric differential system : n X j =1 a ij x j ∂ j − β i • f = 0 , ( i = 1 , . . . , d ) n X j =1 w j x j ∂ j − t∂ t • f = 0 , n Y i =1 ∂ u i i t u n +1 − n Y j =1 ∂ v j j t v n +1 • f = 0 . ( u, v ∈ N n +1 and ˜ Au = ˜ Av )Let I ˜ A be the toric ideal generated by n Y i =1 ∂ u i i t u n +1 − n Y j =1 ∂ v j j t v n +1 ( u, v ∈ N n +1 and ˜ Au = ˜ Av ) (1)in C [ ∂ , . . . , ∂ n , t ]. Since C [ ∂ , . . . , ∂ n , t ] /I ˜ A is an integral domain and t m doesnot belong to the toric ideal, we have I ˜ A = I sat ˜ A = ( I ˜ A : t ∞ ) = { ℓ | t m ℓ ∈ I ˜ A for a non-negative integer m } (2)This fact will be used in the proof of Theorem 2.We note that the matrix ˜ A with w = (1 , . . . ,
1) was introduced in [6] toconstruct vol( A ) convergent series solutions.Throughout this paper, we will use notations and facts shown in [9]. Inparticular, we do not cite original papers for text level well-known facts in thetheory of D-modules. Refer references of [9] as to these original papers.Let a i be the i -th column vector of the matrix A and F ( β, x, t ) the integral F ( β, x, t ) = Z C exp n X i =1 x i t w i s a i ! s − β − ds, s = ( s , . . . , s d ) , β = ( β , . . . , β d ) . he integral F ( β, x, t ) satisfies the modified A -hypergeometric differentialsystem “formally”. We use the word “formally” because, there is no general andrigorous description about the cycle C . However, the integral representationgives an intuitive figure of what are solutions of modified A -hypergeometricsystems. The proof is analogous to [9, 221–222]. We note that if a di = 1 for all i , we also have the following “formal” integral representation F ( β, x, t ) = Z C n X i =1 x i t w i ˜ s ˜ a i ! − β d ˜ s − ˜ β − d ˜ s, ˜ a i = ( a i , . . . , a d − ,i ) T , ˜ s = ( s , . . . , s d − ) , β = ( β , . . . , β d ) . We denote by D the ring of differential operators C h x , . . . , x n , t, ∂ , . . . , ∂ n , ∂ t i .We will regard modified A -hypergeometric system as the left ideal in D . Wewill denote by H A,w ( β ) the left ideal as long as no confusion arises. Theorem 1
1. The left D -module D/H
A,w ( β ) is holonomic.2. The rank of H A,w ( β ) agrees with the holonomic rank of H A ( β ) for any w .Proof . (1) We apply the Laplace transformation with respect to the variable t ( t
7→ − ∂ t ′ , ∂ t t ′ ) for the modified A -hypergeometric system H A,w ( β ). Then,the transformed system is nothing but A -hypergeometric system for the matrix˜ A and the parameter vector ( β , . . . , β n , − ϕ on C n × C ∗ C n × C ∗ ∋ ( y , . . . , y n , s ) ( t w x , . . . , t w n x n , t ) ∈ C n × C ∗ (3)The map ϕ induces a correspondence of differential operators on C n × C ∗ ∂∂y i = t − w i ∂∂x i − s ∂∂s = − t ∂∂t + n X j =1 w n x n ∂∂x i Consider a left ideal H Y in D Y = C h y , . . . , y n , s, ∂ y , . . . , ∂ y n , ∂ s i generated by H A ( β ) and s∂ s . The holonomic rank of D Y /H Y is that of H A ( β ). We cansee that the image of D Y / D Y H Y by the biholomorphic map ϕ on C n × C ∗ is D X / D X H A,w ( β ) by utilizing the correspondence of differential operators. Here, D X and D Y denote the sheaves of differential operators on C n × C ∗ of ( y, s )-space and ( x, t )-space respectively. Since the holonomic rank agrees with themultiplicity of the zero section of the characteristic variety at generic points,the holonomic ranks of the both systems agree [9, pp 28–40]. Q.E.D. orollary 1 rank ( H A ( β )) ≥ vol( A ) Proof . When A has (1 , , . . . ,
1) in its row space ( A is homogeneous), rank ( H A ( β )) ≥ vol( A ) holds [9, Theorem 3.5.1], which is proved by utilizing that H A ( β ) is reg-ular holonomic and by constructing vol( A ) many series solutions. Put w =(1 , , . . . ,
1) in the modified system H ˜ A ( β ). Then, we have rank ( H ˜ A ( β )) ≥ vol( A ). Hence, Theorem 1 gives the conclusion. Q.E.D.Note that the upper semi continuity theorem of holonomic rank of [4] alsogives this result. Example 1
We take A = (1 , β = ( − w = ( − ,
0) (
Airy type integral )[9, p.223]. Define a sequence d m by d = 1 , d m +1 = − (3 m + 1)(3 m + 2)(3 m + 3) m d m The divergent series f ( x ; t ) = ∞ X m =0 (cid:0) d m x − m − x m (cid:1) t m +1 = ∞ X m =0 (cid:18) Γ(3 m + 1)Γ( m + 1) x − m − x m (cid:19) t m +1 (4)is a formal solution of the modified system. Fix a point ( x , x ) = ( a , a ) suchthat a , a = 0. Then this is a Gevrey formal power series solution at ( a , a , t = 0 in the class s = 1 + 2 / sm1.slope , slope ], [1], [2] and the set of the slopes is {− / } . Since 1 / (1 − s ) is the slope,we have constructed a formal power series standing for the slope. A -HypergeometricSystems We will call t = 0 the exceptional hypersurface and we are interested in localanalysis near t = 0. We denote by τ = ( , − ,
1) the weight vector such that t has the weight − ∂ t has the weight 1. We also denote by ˜ A θ,w,β the first( d + 1) Euler operators of the modified A -hypergeometric system.It is easy to see that, for generic w , in τ ( D · I ˜ A ) is generated by monomialsin C [ ∂ , . . . , ∂ n ] and we will regard it as a monomial ideal in this commutativering. Theorem 2
For generic β and w , we have in ( , − , ( H A,w ( β )) = D · in τ ( D · I ˜ A ) + D · ˜ A θ,w,β (5) roof . The proof is analogous to [9, Theorem 3.1.3]. Let s = ( s , . . . , s d ) bea vector of new indeterminates. Consider the algebra D [ s ] = C h x , . . . , x n , t, ∂ , . . . , ∂ n , ∂ t , s , . . . , s d i and its homogenized Weyl algebra by h D [ s ] h . Let H be the left ideal in D [ s ] h generated by ˜ A θ,w,s and the homogenization of I ˜ A . We define a partial order > τ on monomials in D [ s ] by s a x b ∂ c t d ∂ et > τ s a ′ x b ′ ∂ c ′ t d ′ ∂ e ′ t ⇔ − d + e > − d ′ + e ′ , or − d + e = − d ′ + e ′ and ( a, e, d ) > lex ( a ′ , e ′ , d ′ )We refine this partial order by any monomial order and define orders < in D [ s ].(This order on D [ s ] is extended to the order in the homogenized Weyl algebraand D [ s ] h as in [9, Chapter 1].)Let G be the reduced Gr¨obner basis of the homogenized binomial ideal I ˜ A in D [ s ] h with respect to the order < . Note that the reduced Gr¨obner basis consistsof elements of the form ∂ u h p − ∂ v t v n +1 h p ′ , v n +1 > w is generic and I ˜ A is saturated with respect to t . Note that either p = 0 or p ′ = 0 holds.We will show that G and ˜ A hθ,w,s is a Grobner basis G ′ with respect to < in D [ s ] h . This fact can be shown by checking the S-pair criterion in D [ s ] h . It iseasy to see that sp ( θ t − X w j θ j , s i − X a ij θ j ) → G ′ sp ( s k − X a kj θ j , s i − X a ij θ j ) → G ′ p > p ′ = 0. sp ( ∂ u h p − ∂ v t v n +1 , s i − X a ij θ j )= s i ( ∂ u h p − ∂ v t v n +1 ) − ∂ u h p ( s i − X a ij θ j ))= − s i ∂ v t v n +1 + ∂ u h p X a ij θ j = − s i ∂ v t v n +1 + (cid:16)X a ij θ j (cid:17) ∂ u h p + (cid:16)X a ij u j (cid:17) ∂ u h p since ∂ u h p > ∂ v t v n +1 we may rewrite it as= − s i ∂ v t v n +1 + (cid:16)X a ij θ j (cid:17) ( ∂ u h p − ∂ v t v n +1 ) + (cid:16)X a ij θ j (cid:17) ∂ v t v n +1 + (cid:16)X a ij u j (cid:17) ∂ u h p = (cid:16)X a ij θ j (cid:17) ( ∂ u h p − ∂ v t v n +1 ) + ∂ v t v n +1 (cid:16)X a ij θ j − X a ij v j − s i (cid:17) + (cid:16)X a ij u j (cid:17) ∂ u h p since P a ij u j = P a ij v j = (cid:16)X a ij θ j (cid:17) ( ∂ u h p − ∂ v t v n +1 ) + ∂ v t v n +1 (cid:16)X a ij θ j − s i (cid:17) + (cid:16)X a ij u j (cid:17) ( ∂ u h p − ∂ v t v n +1 ) → G ′ p = 0, and p ′ > sp ( ∂ u h p − ∂ v t v n +1 , θ t − X a ij θ j )= − θ t ∂ v t v n +1 + h p ∂ u X w j θ j = − θ t ∂ v t v n +1 + (cid:16)X w j θ j + X w j u j (cid:17) h p ∂ u = − θ t ∂ v t v n +1 + (cid:16)X w j θ j + X w j u j (cid:17) ( h p ∂ u − ∂ v t v n +1 )+ (cid:16)X w j θ j + X w j u j (cid:17) ∂ v t v n +1 = − θ t ∂ v t v n +1 + (cid:16)X w j θ j + X w j u j (cid:17) ( h p ∂ u − ∂ v t v n +1 )+ ∂ v t v n +1 (cid:16)X w j θ j + X w j u j − X w j v j (cid:17) = (cid:16)X w j θ j + X w j u j (cid:17) ( h p ∂ u − ∂ v t v n +1 )+ ∂ v t v n +1 (cid:16)X w j θ j + X w j u j − X w j v j − θ t − v n +1 (cid:17) = (cid:16)X w j θ j + X w j u j (cid:17) (cid:16) h p ∂ u − ∂ v t v n +1 (cid:17) + ∂ v t v n +1 (cid:16)X w j θ j − θ t (cid:17) → G ′ t = 0 We fix generic w . Let M be the monomial ideal in τ ( I ˜ A ) in C [ ∂ , . . . , ∂ n ]. Thetop dimensional standard pairs are denoted by T ( M ) [9, p.112] and β ( ∂ β ,σ ) isthe zero point in C n of the distraction of M and Aθ − β associated to thestandard pair ( ∂ β , σ ). Theorem 3
Let β and w both be generic. Then, the indicial polynomial ( b -function) of H A,w ( β ) along t = 0 is X ( ∂ β ,σ ) ∈T ( M ) ( s − w · β ( ∂ β ,σ ) ) (6) If T ( M ) is the empty set, the indicial polynomial is .Proof . Under Theorem 2, the proof is analogous to [9, p.198, Proposition5.1.9].If the indicial polynomial is not zero and the difference of roots are notintegral, we can construct formal series solution of the form t e ∞ X k =0 c k ( x ) t k , c k ∈ C [1 /x , . . . , /x n , x , . . . , x n ] (7)here e is a root of the indicial polynomial and t e c ( x ) is a solution of theinitial system in ( , − , ( H A,w ( β )). If the indicial polynomial is zero, there isno formal series solution of the form above. Example 2 (Continuation of Example 1). Note that in τ ( I ˜ A ) = h ∂ i . Thedistraction [9, p.68] of in τ ( H A,w ( β )) is generated by θ , θ + 3 θ + 1 , − θ − θ t .Therefore, the set of zero points are { ( − , , } . Then, the indicial polynomialis s −
1. The formal solution (4) stands for the root s = 1. Example 3
Consider the modified hypergeometric system for A = ( − , , β = (1 / w = ( − , − , Bessel function in two variables calledby Kimura and Okamoto [5]. Although it is a side story in view of this paper,we want to note that a 3-D Graph of a solution of this system can be seen at . You will beable to see waves in two directions.The indicial polynomial is 0, because I ˜ A ∋ − ∂ ∂ . Then, there exists noseries solution of the form (7). Incidentally, the set of the slopes along t = 0 at x = (2 , ,
1) is equal to {− , − / } . The values are obtained by our program[7]. Let us change w into w = (3 , , { ( − / , , , (0 , , , (0 , , − / } (which are obtained by computing the pri-mary decomposition of the ideal generated by the distraction of in τ ( I ˜ A ) and˜ A θ,w,β ) and the indicial polynomial is ( s − / s + 1 / s − /
4) (we use the
Risa/Asir command generic_bfct ). In this case, the generic condition forthe Theorem 3 satisfied and the equality (6) also gives the same answer. In-cidentally, the local monodromy group of the local solutions around t = 0 isgenerated by diag( − , exp( π √− / , − exp( π √− / t = 0 at x = (2 , ,
1) is empty.It is an interesting open problem to construct rank many series solutions interms of formal puiseux series and exponential functions along t = 0. Acknowledgements : This work was motivated by comments by Bernd Sturm-fels to the joint work with Francisco Castro-Jimenez on slopes for A -hypergeometricsystem [2]. He said that “can you construct series solutions standing for theslopes”? We realized that the original A -hypergeometric system has few clas-sical solutions standing for slopes for some examples. The author introduced amodified system, which seems to be easier to analyze than the original systemand may be a step to study the original A -hypergeometric system, when Castrostayed in Japan in the spring of 2006. In fact, our Example 1 gives an exam-ple to his question. However, we are far from a complete answer. The authorthanks to all comments and discussions with them.The author also thanks to Christine Berkesch who made me some questionson the first versioin of this paper, which yields a substantial improvement in thesecond version.inally, Go Okuyama made a question on the lower bound of the rank of A -hypergeometric system. The Corollary 1 is an answer to his question. References [1] A.Assi, F.J.Castro-Jim´enez, M.Granger, How to calculate the slopes of a D -module. Compositio Mathematica , (1996) 107–123.[2] Francisco Castro-Jimenez and Nobuki Takayama, Singularities of the Hy-pergeometric System associated with a Monomial Curve. Transaction ofthe American Mathematica Society, (2003), 3761–3775.[3] Gel’fand, I.M., Zelevinsky, A.V., Kapranov, M.M. (1989): Hypergeometricfunctions and toral manifolds. Functional Analysis and its Applications ,94–106.[4] L. F. Matusevich, E. Miller, and U. Walther, Homological Methods forHypergeometric Families, Journal of American Mathematical Society 18(2005), 919–941. math.AG/0406383 .[5] Toshinori Oaku, Yoshinao Shiraki, Nobuki Takayama, Algebraic Algo-rithms for D-modules and Numerical Analysis, Z.M.Li, W.Sit (editors),Computer Mathematics, Proceedings of the sixth Asian symposium, 23–39, World scientific, 2003.[6] K.Ohara, N.Takayama, Holonomic rank of A -hypergeometric differential-difference equations, math.CA/0706.2706v1 [7] OpenXM, a project to integrate mathematical software systems, 1998–2007, , download the full package and startOpenXM/Risa/Asir.[8] B.Sturmfels, Asymptotic analysis of toric ideals. Mem. Fac. Sci. KyushuUniv. Ser. A 46 (1992), 217–228.[9] M.Saito, B.Sturmfels, and N.Takayama, Gr¨obner Deformations of Hyper-geometric Differential Equations , Springer, 2000.[10] M.Schulze, U.Walther, Slopes of Hypergeometric Systems along CoordinateVarieties,, Springer, 2000.[10] M.Schulze, U.Walther, Slopes of Hypergeometric Systems along CoordinateVarieties,