Modified A-hypergeometric Systems
Nobuki Takayama, Department of Mathematics, Kobe University
June 30, 2007, January 21, 2008 revised, March 6, 2008 revised
Abstract We introduce a modification of A-hypergeometric systems (GKZ systems) by applying a change of variables for Gröbner deformations and study its
Gröbner basis and indicial polynomial along the exceptional hypersurface.
Key words: A-hypergeometric system, GKZ system, indicial polynomial, Gröbner
basis, modifed A-hypergeometric system
MSC number: 33C70

1Introduction

An A-hypergeometric system is a system of multidimensional hypergeometricpartial differential equations defined by a matrix A and a set of parameters β.
It is defined on (y1 , . . . , yn ) space. We will denote such a system by HA (β).
By specializing parameters and some independent variables, we obtain classicalhypergeometric differential equations in one or several variables. Ever since thestudy of Gel’fand, Zelevinsky, and Kapranov [3], studies of A-hypergeometricsystems (GKZ systems) have attracted the interest of many mathematicians,
who wish to understand hypergeometric differential equations in a general way.
We refer the reader to the book [11] for a review of important studies on Ahypergeometric systems up to 2000, and the recent papers [5] and [12] and theirreferences for recent advances. We also note that these studies have interacted
fruitfully with the frontiers of computational commutative algebra and computational D-modules (where D denotes the ring of differential operators withpolynomial coefficients).
In this short paper, we introduce a modified version of A-hypergeometricsystems and indicate a first step for studying them. The original system isdefined on y = (y1 , . . . , yn ) space, whereas the modified system is definedon (t, x1 , . . . , xn ) space with an additional variable t. Let us sketch our idea
for introducing the modified system. We consider the direct sum of the Ahypergeometric D-module on y space and the D-module D/D · s∂s on s-space.
For the weight vector w ∈ Zn , the original system restricted on Cn × C∗ istransformed into the modified system on (t, x) space by the map
Cn × C∗ 3 (y1 , . . . , yn , s) 7→ (x1 , . . . , xn , t) ∈ Cn × C∗
where

(x1 , . . . , xn , t) = (s−w1 y1 , . . . , s−wn yn , s)1and(y1 , . . . , yn , s) = (tw1 x1 , . . . , twn xn , t)(see [9] and [11] on this transformation). The transformed system can be naturally extended on Cn+1 . The system is firstly defined on t 6= 0 and is extended
to the whole space Cn+1 including t = 0. Intuitively speaking, the hypersurfacet = 0 is analogous to the exceptional hypersurface of a blowing-up operation.
We rigorously define the modified system in the next section.
We are interested in how solutions behave near the exceptional hypersurfacet = 0. We will study the indicial polynomial along t = 0 as a first step toperforming local and global analyses of the modified system.

2

Definition and Holonomic Rank of Modified
A-hypergeometric systems

Let A = (aij ) be a d×n-matrix whose elements are integers and w = (w1 , . . . , wn )a vector of integers. We suppose that the set of the column vectors of A spans
Zd . Set
a11 · · · a1n 0···0Ã = 
.
adn · · · adn 0
w1 · · · wn 1
Definition 1 We call the following system of differential equations HA,w (β) amodified A-hypergeometric system:


n
X

aij xj ∂j − βi  • f = 0,(i = 1, . . . , d)





j=1
n
X



wj xj ∂j − t∂t  • f

j=1
n
Y

n
Y

∂iui tun+1 −

i=1

= 0,


v

∂j j tvn+1  • f= 0.

(u, v ∈ Nn+1
run over all u, v such that Ãu = Ãv)
0

j=1

Here, N0 = {0, 1, 2, . . .} and β = (β1 , . . . , βd ) ∈ Cd are parameters.
Let IÃ be the toric ideal generated bynY
i=1∂iui tun+1 −nYv∂j j tvn+1(u, v ∈ Nn+1
run over all u, v such that Ãu = Ãv)
0

j=1

(1)
in C[∂1 , . . . , ∂n , t]. Since C[∂1 , . . . , ∂n , t]/IÃ is an integral domain and tm does
not belong to the toric ideal, we havesatIÃ = IÃ:= (IÃ : t∞ ) := {` | tm ` ∈ IÃ for a non-negative integer m}(2)This fact will be used in the proof of Theorem 2.
We note that the matrix Ã with w = (1, . . . , 1) was introduced in [7] toconstruct a fundamental set of convergent series solutions.
Throughout this paper, we will use facts shown in [11]. However, they
are limited to only a small part of this book and this paper is written to beunderstandable with an elementary knowledge on Gröbner bases in the ring ofdifferential operators and basics on D-modules.
Before starting an algebraic discussion about the modified A-hypergeometricsystem, we informally discuss a bit about an integral representation of the modified system to explain about a relation between modified systems and classicalhypergeometric functions. Let ai be the i-th column vector of the matrix A andF (β, x, t) the integral
Ã n
!
Z
X
F (β, x, t) =exp
xi twi sai s−β−1 ds,
s = (s1 , . . . , sd ), β = (β1 , . . . , βd ).
C

i=1

Qd

Qd
aji
−β −1
−β−1
= j=1 sj j , ds = ds1 · · · dsd .
j=1 sj , sHere, sai =The integral
F (β, x, t) satisfies the modified A-hypergeometric system “formally”, which canbe shown by integration by parts (see, e.g., [11, 221–222]). We use the word“formally”, because we have no general and rigorous description about the cycle
C except the case of d = 1 [4]. However, the integral representation gives
an intuitive description of what are solutions of modified A-hypergeometricsystems. We note that if adi = 1 for all i, we also have the following “formal”
integral representation
!−βd
Z ÃX
n
xi twi s̃ãis̃−β̃−1 ds̃,F (β, x, t) =Ci=1

ãi = (a1i , . . . , ad−1,i )T , s̃ = (s1 , . . . , sd−1 ), β̃ = (β1 , . . . , βd−1 ).
We denote by D the ring of differential operators Chx1 , . . . , xn , t, ∂1 , . . . , ∂n , ∂t i.
We will regard the modified A-hypergeometric system as the left ideal in D. We
will denote by HA,w (β) this left ideal as long as no confusion arises and call it
the modified A-hypergeometric ideal.
A set of generators of HA,w (β) can be computed by computing a set ofgenerators of the toric ideal IÃ . An algorithm of computing a set of generatorsof the toric ideal is given in the book [10, Algorithm 4.5].
Here is a log of a session with the computer algebra system kan/sm1 [8] toobtain a set of generators of HA,w (β) for A = (−1, 1, 2), w = (−2, −1, 0) and
β = (0).
(cohom.sm1) run[ [[-1,1,2]], [-2,-1,0], [0]] mgkz ::[-x1*Dx1+x2*Dx2+2*x3*Dx3, -2*x1*Dx1-x2*Dx2-x4*Dx4,
x4^2*Dx2^2-Dx3,-x4^3*Dx1*Dx2+1,x4*Dx1*Dx3-Dx2,-x4*Dx2^3+Dx1*Dx3^2,-Dx1^2*Dx3^3+Dx2^4]

The command mgkz outputs a set of generators of HA,w (β). Here, x1, x2, x3,
x4 stand for x1 , x2 , x3 and t respectively and Dxi stands for ∂i .

The rank of HA,w (β) is the dimension of the C-vector space of the classicalsolutions of HA,w (β) and is denoted by rank (HA,w (β)) [11, p.31].
Theorem 1

1. The left D-module D/HA,w (β) is holonomic.

2. The rank of HA,w (β) agrees with the rank of HA (β) for any w.
Proof. (1) We apply the Laplace transformation with respect to the variablet ( t 7→ −∂t0 , ∂t 7→ t0 ) for the modified A-hypergeometric ideal HA,w (β). Then,the transformed system is nothing but A-hypergeometric ideal for the matrixÃ and the parameter vector (β1 , . . . , βn , −1). It is known that the transformed
system is holonomic, then the original system is also holonomic by showing
the Hilbert polynomials with respect to the Bernstein filtration of each systemagree.
(2) We consider the biholomorphic map ϕ on Cn × C∗
Cn ×C∗ 3 (y1 , . . . , yn , s) 7→ (s−w1 y1 , . . . , s−wn yn , s) = (x1 , . . . , xn , t) ∈ Cn ×C∗(3)
The map ϕ induces a correspondence of differential operators on Cn × C∗∂∂yi

= t−wi

∂
∂s

= −t

−s

∂
∂xi

n
X
∂
∂
+
wn xn
∂t j=1∂xi

Consider a left ideal HY in DY = Chy1 , . . . , yn , s, ∂y1 , . . . , ∂yn , ∂s i generated by
HA (β) and s∂s . The holonomic rank of DY /HY is that of HA (β). We cansee that the image of DY /DY HY by the biholomorphic map ϕ on Cn × C∗ is
DX /DX HA,w (β) by utilizing the correspondence of differential operators. Here,DX and DY denote the sheaves of differential operators on Cn × 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 [11, pp 28–40]. Q.E.D.
We denote by vol(A) the normalized volume of the convex hull of the column
vectors of matrix A and the origin [11, p.50].
Corollary 1 rank (HA (β)) ≥ vol(A)Proof. When A has (1, 1, . . . , 1) in its row space (A is homogeneous), rank (HA (β)) ≥vol(A) holds [11, Theorem 3.5.1], which is proved by utilizing that HA (β) is
regular holonomic and by constructing vol(A) many series solutions. Put w =(1, 1, . . . , 1) in the modified system HA,w (β). Then, we have rank (HA,w (β)) ≥vol(A). Hence, Theorem 1 gives the conclusion. Q.E.D.
Note that the upper semi continuity theorem of holonomic rank of [5] also
gives this result.

The aim of this paper is to determine the indicial polynomial of HA,w (β)
along t = 0 and construct formal series solutions standing for roots of the indicial
polynomial. Before starting a systematic discussion, we show one example ofconstruction of a formal series solution.
Example 1 We take A = (1, 3), β = (−1), and w = (−1, 0) (Airy type integral)
3m+1[11, p.223]. The monomial x−3m−1xmsatisfies the two first order equa2 t
1tions of the Definition 1. Then, we can expect a superposition of monomialsof thisform is a formal solution. The ideal IÃ is generated by ∂2 − t3 ∂13 . PutP∞
3m+1
f = m=0 dm x1−3m−1 xm
and determine dm by utilizing ∂2 − t3 ∂13 . Then,2 t
we can see that dm satisfies the following recurrence−(3m + 1)(3m + 2)(3m + 3)dmmd0 = 1, dm+1 =The divergent seriesf (x; t) =
=

∞
X
¡

¢ 3m+1
dm x1−3m−1 xm
2 t

m=0
∞ µ
X

(−1)m

m=0

¶
Γ(3m + 1) −3m−1 m 3m+1
x1
x2 t
Γ(m + 1)

(4)

is a formal solution of the modified system. Fix a point (x1 , x2 ) = (a1 , a2 ) suchthat a1 , a2 6= 0. Then this is a Gevrey formal power series solution at (a1 , a2 , 0)along t = 0 in the class s = 1 + 2/3 from the definition of Gevrey series.
The slope of a holonomic system is a set of numbers each of which stands fordiscontinuity of a one parameter family of micro-characteristic varieties [1]. The
slope of this system can be computed by our software [8, command sm1.slope,slope], which uses algorithms in [1], [2] and the set of the slopes is {−3/2}.
Since 1/(1 − s) agrees with −3/2, we have constructed a formal power series
standing for the slope −3/2.

3Gröbner Bases of Modified A-Hypergeometric
Systems

We will call t = 0 the exceptional hypersurface and we are interested in local
analysis near t = 0. We denote byτ = (0, −1; 0, 1)
the weight vector such that t has the weight −1 and ∂t has the weight 1. θi
is xi ∂i and θt is t∂t . We denote by Ãθ,w,β the set of the first (d + 1) Euler
operators of the modified A-hypergeometric system. For ` ∈ D, inτ (`) is thesum of the maximal τ -order terms in `. For a left ideal J in D, inτ (J) is theleft ideal in D generated by inτ (`), ` ∈ J [11, p.4].
It is easy to see that, for generic w, inτ (D·IÃ ) can be generated by monomialsin C[∂1 , . . . , ∂n ].

Theorem 2 For generic β and w, we haveinτ (HA,w (β)) = D · inτ (D · IÃ ) + D · Ãθ,w,β(5)

Proof. Let G 00 be a Gröbner basis of the left ideal HA,w (β) for an orderwhich refines the partial order determined by the weight vector τ . Then, theset {inτ (g) | g ∈ G 00 } generates inτ (HA,w (β)) [11, Theorem 1.1.6]. The proof
is performed by applying the Buchberger algorithm of computing the Gröbner
basis G 00 .
Let s = (s1 , . . . , sd ) be a vector of new indeterminates. Consider the algebra
D[s] = Chx1 , . . . , xn , t, ∂1 , . . . , ∂n , ∂t , s1 , . . . , sd i
and its homogenized Weyl algebra by hD[s]h = Chx1 , . . . , xn , t, ∂1 , . . . , ∂n , ∂t , s1 , . . . , sd , hi
where ∂i xj = xj ∂i + h2 δij and h commutes with other variables. Let H be
the left ideal in D[s]h generated by Ãθ,w,s2 and the homogenization of IÃ . We
define a partial order >τ on monomials in D[s] by
0

0

0

0

sa xb ∂ c td ∂te >τ sa xb ∂ c td ∂te0⇔−d + e > −d0 + e0 , or−d + e = −d0 + e0 and (a, e, d) >lex (a0 , e0 , d0 )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 algebra
and D[s]h as in [11, Section 1.2].)
Let G be the reduced Gröbner basis of the homogenized binomial ideal IÃ in
D[s]h with respect to the order <. Note that the reduced Gröbner basis consists
0of elements of the form ∂ u hp − ∂ v tvn+1 hp , vn+1 > 0 because w is generic andIÃ is saturated with respect to t. Note that either p = 0 or p0 = 0 holds.
We will show that G and Ãθ,w,s2 is a Gröbner basis G 0 with respect to < inD[s]h . This fact can be shown by checking the S-pair criterion in D[s]h . It iseasy to see thatX
X
sp(θt −
wj θj , s2i −aij θj ) →G 0 0XX
akj θj , s2i −aij θj ) →G 0 0
sp(s2k −We assume p > 0 and p0 = 0.
sp(∂ u hp − ∂ v tvn+1 , s2i −
=
=
=

X

aij θj )X
s2i (∂ u hp − ∂ v tvn+1 ) − ∂ u hp (s2i −aij θj ))X−s2i ∂ v tvn+1 + ∂ u hp
aij θj³X´³X´−s2i ∂ v tvn+1 +aij θj ∂ u hp +aij uj ∂ u hp

since ∂ u hp > ∂ v tvn+1 we may rewrite it as

=

³X
´
³X
´
³X
´
−s2i ∂ v tvn+1 +aij θj (∂ u hp − ∂ v tvn+1 ) +aij θj ∂ v tvn+1 +aij uj ∂ u hp
³X
´
³X
´ ³X
´
X
aij θj (∂ u hp − ∂ v tvn+1 ) + ∂ v tvn+1
aij θj −aij vj − s2i +aij uj ∂ u hp
P
P
since
aij uj = aij vj
³X
´ ³X
´
³X
´
aij θj − s2i +aij uj (∂ u hp − ∂ v tvn+1 )aij θj (∂ u hp − ∂ v tvn+1 ) + ∂ v tvn+1

→G 0

0

=
=

The case p = 0, and p0 > 0 can be shown analogously.
The final case we have to check is thatXsp(∂ u hp − ∂ v tvn+1 , θt −wj θj )X=−θt ∂ v tvn+1 + hp ∂ u
wj θj
³X
´
X
=
−θt ∂ v tvn+1 +wj θj +
wj uj hp ∂ u³X´
X=−θt ∂ v tvn+1 +wj θj +
wj uj (hp ∂ u − ∂ v tvn+1 )´³XX
wj θj +
wj uj ∂ v tvn+1+´³X
X
wj θj +
wj uj (hp ∂ u − ∂ v tvn+1 )=−θt ∂ v tvn+1 +´³XXX
wj θj +
wj uj −wj vj+∂ v tvn+1³X´
X=
wj θj +
wj uj (hp ∂ u − ∂ v tvn+1 )³X´
XX+∂ v tvn+1
wj θj +
wj uj −
wj vj − θt − vn+1
³X
´³
´
´
³X
X
=
wj θj +
wj uj hp ∂ u − ∂ v tvn+1 + ∂ v tvn+1
wj θj − θt
→G 00Specializing s to a vector of generic numbers β and h to 1 in G 0 , we have aGröbner basis of HA,w (β) with respect to τ . The correctness proof of this fact
and the rest of the proof are analogous to [11, Theorem 3.1.3, p.106]. Q.E.D.

4

Indicial Polynomial along t = 0

The indicial polynomial of HA,w (β) along t = 0 is the monic generator b(θt ) of
the principal ideal inτ (HA,w (β)) ∩ C[θt ]. We fix a generic weight vector w.
Since w is generic, inτ (IÃ ) can be generated by monomials in C[∂1 , . . . , ∂n ]. Let
M be the monomial ideal generated by these monomials in C[∂1 , . . . , ∂n ]. Theβ
top dimensional standard pairs are denoted by T (M ) [11, p.112] and β (∂ ,σ) isthe zero point in Cn of the distraction D · M ∩ C[θ1 , . . . , θn ] of M and Aθ − βassociated to the standard pair (∂ β , σ) [11, p.68]. Here, the ring D in this paragraph is Chx1 , . . . , xn , ∂1 , . . . , ∂n i and we regard the sub ring C[θ1 , . . . , θn ] ⊂ D
as a commutative polynomial ring in variables θi .

Theorem 3 Let both of β and w be generic. Then, the indicial polynomial b(s)of HA,w (β) along t = 0 isXβ(s − w · β (∂ ,σ) )(6)(∂ β ,σ)∈T (M )If T (M ) is the empty set, the indicial polynomial is 0.
Proof. Under Theorem 2, the proof is analogous to [11, p.198, Proposition
5.1.9].
If the indicial polynomial is not constant and the difference of roots are notintegral, we can construct formal series solution of the form
e

t

∞
X

ck (x)tk ,ck ∈ C[1/x1 , . . . , 1/xn , x1 , . . . , xn ]

(7)

k=0

where e is a root of the indicial polynomial and te c0 (x) is a solution of the initialsystem in(0,−1;0,1) (HA,w (β)). If the indicial polynomial is a constant, there is
no formal series solution of the form above.
Example 2 (Continuation of Example 1). Note that IÃ is generated by ∂2 −t3 ∂13 and inτ (IÃ ) = h∂2 i. The distraction [11, p.68] of M = h∂2 i is θ2 . The zero
set of the ideal generated by θ2 and Aθ − β = {θ1 + 3θ2 + 1} is {(−1, 0)}. Then,the indicial polynomial is s − w · (−1, 0) = s − 1. The formal solution (4) stands
for the root s = 1.
Example 3 Consider the modified hypergeometric system for A = (−1, 1, 2),
β = (1/2), w = (−2, −1, 0). This is the Bessel function in two variables calledby Kimura and Okamoto (see, e.g., [6]).
The indicial polynomial is 1, because IÃ 3 1 − ∂12 ∂3 . Then, there exists noseries solution of the form (7). Incidentally, the set of the slopes along t = 0 at
x = (2, 2, 1) is equal to {−2, −3/2}. The values are obtained by our software[8].
Let us change w into w = (3, 2, 1). The initial ideal inτ (IÃ ) is generated by the set {∂1 ∂3 , ∂22 , ∂1 ∂2 } and the distraction of it is generated by θ1 θ3 ,
θ2 (θ2 − 1), θ1 θ2 . The set of zero points of the distraction and Aθ − β is{(−1/2, 0, 0), (0, 0, 1/4), (0, 1, −1/4)} which is obtained by computing the primary decomposition of the ideal generated by the distraction and Aθ − β by thecomputer algebra system Risa/Asir [8] as[0] load("gr"); load("primdec");[107] [182] G=[x1*x3 , x2*(x2-1) , x1*x2 , -2*x1+2*x2+4*x3-1];[x3*x1,x2^2-x2,x2*x1,-2*x1+4*x3+2*x2-1][183] primadec(G,[x1,x2,x3]);[[[x3,x2,2*x1+1],[x3,x2,2*x1+1]],[[4*x3-1,x2,x1],[4*x3-1,x2,x1]],[[4*x3+1,x2-1,x1],[4*x3+1,x2-1,x1]]]

In this case, the generic condition for the Theorem 3 satisfied and the formula(6) gives the indicial polynomial (s − 3/2)(s + 1/4)(s − 7/4). We may use theRisa/Asir command generic_bfct to compute the indicial polynomial, but
it uses a lot of memory and computation time when A is relatively large matrix.
Our formula (6) can be applied to larger A. The rank of this system is 3. The
set of the slopes along t = 0 at x = (2, 2, 1) is empty.
Although it is a side story in view of this paper, we want to note that a
3-dimensional graph of a solution of this system can be seen at
http://www.math.kobe-u.ac.jp/HOME/taka/test-bess2m.html. We can see
waves in two directions. This graph is drawn by series solutions and the RungeKutta method [6].
The number of solutions of the form (7) is less than the rank in general. It
is an interesting open problem to construct rank many series solutions in termsof formal puiseux series and exponential functions along t = 0.
Acknowledgments: This work was motivated by comments by Bernd Sturmfels in the joint study with Francisco Castro-Jimenez on slopes for A-hypergeometricsystems [2]. He asked “Is it possible to construct series solutions representing theslopes”? We realized that the original A-hypergeometric system has few classical solutions representing slopes for some examples. The author introduced amodified system, which appears to be easier to analyze than the original system
and may be the first step to studying the original A-hypergeometric system.
We did this work when Castro stayed in Japan in the spring of 2006. In fact,
our Example 1 is an example to his question. However, we are still far fromobtaining a complete answer. The author would like to thank both of them for
all their comments and discussions.
The author also thanks Christine Berkesch who checked the first draft ofthis paper, helping to improve it substantially.
Finally, Go Okuyama posed a question on the lower bound of the rank of an
A-hypergeometric system. The Corollary 1 is the answer to his question.

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Nobuki Takayama
Department of Mathematics, Kobe University
Rokko, Kobe, 657-8501, Japan
takayama@math.kobe-u.ac.jp