Table Of ContentExponential growth of rank jumps for
A–hypergeometric systems.
2
1 Mar´ıa-Cruz Fern´andez-Fern´andez
0 ∗
2 Departamento de A´lgebra
n
Universidad de Sevilla
a
J
4 January 24, 2012
2
]
G
A Abstract
.
h The dimension of the space of holomorphic solutions at nonsingular
t points (also called the holonomic rank) of a A–hypergeometric system
a
m M (β)is knowntobeboundedabove by22dvol(A) [SST00], wheredisthe
A
[ rank of the matrix A and vol(A) is its normalized volume. This boundwas
thought to be very vast because it is exponential on d. Indeed, all the ex-
1
v amples we have foundin theliterature verify that rank(MA(β)) < 2vol(A).
0 We construct here, in a very elementary way, some families of matri-
9
ces A Zd×n and parameter vectors β Cd, d 2, such that
0 (d) (d)
∈ ∈ ≥
5 rank(M (β )) advol(A ) for certain a > 1.
. A(d) (d) ≥ (d)
1
0
2 1 Introduction
1
:
v Let A = (a ) = (a a a ) be a full rank matrix with columns a Zd and
i ij 1 2 n j
X ··· ∈
d n. Following Gel’fand, Graev, Kapranov and Zelevinsky (see [GGZ87] and
ar [G≤ZK89]) we can define the A–hypergeometric system with parameter β Cd as
∈
the left ideal H (β) of the Weyl algebra D = C[x ,...,x ] ∂ ,...,∂ generated
A 1 n 1 n
h i
by the following set of differential operators:
(cid:3) := ( ∂ui) ( ∂−ui) for all u Zn such that Au = 0 (1)
u i − i ∈
i:Yui>0 i:Yui<0
and
n
E β := a x ∂ β for i = 1,...,d (2)
i i ij j j i
− −
Xj=1
∗
PartiallysupportedbyMTM2010-19336andFEDER,andJuntadeAndaluc´ıaundergrants
FQM-5849,FQM333. E.mail address: [email protected]
1
The operators given in (1) generate the so-called toric ideal I C[∂ ,...,∂ ]
A 1 n
⊆
associated with A and the d operators given in (2) are called the Euler operators
associated with the pair (A,β). The hypergeometric D–module associated with
the pair (A,β) is the quotient M (β) = D/DH (β). It is a holonomic D–module
A A
for any pair (A,β) as above (see [GZK89], [Ado94]). In particular, the space of
holomorphic solutions of M (β) at a nonsingular point has finite dimension. This
A
dimension or, equivalently, the holonomic rank of M (β) equals the normalized
A
volume volZA(A) of the matrix A (see (3)) when either IA is Cohen-Macaulay or
β is generic (see [GZK89], [Ado94], [SST00]).
The first example of a pair (A,β) for which rank(MA(β)) > volZA(A) was
described in [ST98] (see Example 2.5). A complete description of the case d = 2
appears in [CDD99], revealing that in this case the rank of M (β) can be only
A
volZA(A) (the generic value) or volZA(A)+1 (the exceptional value).
Ingeneralitisknownthatrank(MA(β)) volZA(A)forallβ [SST00,MMW05].
≥
In fact, it is proved in [MMW05] that the map β Cd rank(M (β)) is up-
A
∈ 7→
per semi–continuous in the Zarisky topology and they also provide an explicit
description of the exceptional set
ε(A) = β Cd : rank(MA(β)) > volZA(A)
{ ∈ }
that turns out to be an affine subspace arrangement with codimension at least 2.
Previous descriptions of the exceptional set in particular cases appear in [CDD99,
Mat01, Sai02, Mat03].
If for a fixed matrix A we have that jA(β) = rank(MA(β)) volZA(A) > 0
−
then it said that the A–hypergeometric system has a rank jump of j (β) at β or
A
that β is a rank jumping parameter for A.
The paper [MW07] provides the first family of hypergeometric systems with
rank jump greater than 2. Indeed, they construct a family of pairs (A ,β )
(d) (d)
with A Zd×2d and β Cd such that j (β ) = d 1. However, for this
(d) ∈ (d) ∈ A(d) (d) −
family volZA(d)(A(d)) = d+2 and thus
rank(MA(d)(β(d))) = 2 3 < 2
volZA(d)(A(d)) − d+2
More recently, in [Ber11] a general combinatorial formula is provided for the
rank jump j (β) of the A–hypergeometric system at a given β. However, the
A
formula is very complicated and, in fact, all the examples included in [Ber11]
verify that rank(MA(β)) < 2volZA(A) as well. Previous computations of jA(β) in
particular cases appear for example in [CDD99], [Sai02], [Oku06].
In the case when the toric ideal is standard homogeneous, the following upper
bound for the holonomic rank of a hypergeometric system is proved in [SST00]:
rank(MA(β)) 22dvolZA(A)
≤
2
However, it is mentioned in [SST00, p. 159] that this upper bound is most
likely far from optimal and that it would be desirable to know whether the ratio
rank(MA(β))/volZA(A) can be bounded above by some polynomial function in d.
Here we provide a very elementary construction of some families of hypergeomet-
ric systems for which the ratio rank(MA(β))/volZA(A) is exponential on d, giving
a negative answer to this last question.
Moreover, for one of the families constructed the dimension of Laurent poly-
nomial solutions islower thanthe rankjump(see Remark2.8). This isin contrast
with the general observation in the examples found in the literature (see for ex-
ample [MW07]).
I am grateful to Christine Berkesch for many helpful conversations about her
paper [Ber11].
2 Construction of the examples.
Recall that the normalized volume of a full rank matrix A Zd×n is given by
∈
vol (∆ )
Rd A
volZA(A) = d! [Zd : ZA] (3)
where [Zd : ZA] is the index of the subgroup ZA := n Za Zd, ∆ is the
i=1 i ⊆ A
convex hull of the columns of A and the origin in Rd aPnd volRd(∆A) denotes the
Euclidean volume of the polytope ∆ .
A
Let us also recall that thedirect sumof two matrices A Zd1×n1,A Zd2×n2
1 2
∈ ∈
is the following (d +d ) (n +n ) matrix:
1 2 1 2
×
A 0
A A = 1 d1×n2
1 ⊕ 2 (cid:18) 0d2×n1 A2 (cid:19)
where 0 denotes the d n zero matrix.
d×n
×
The following two lemmas are easy to prove.
Lemma 2.1. If A is the direct sum of two matrices A Zd1×n1,A Zd2×n2
1 2
∈ ∈
then volZA(A) = volZA1(A1)·volZA2(A2).
Lemma 2.2. Let A Zdi×ni be full rank matrices, d n , and β Cdi
i i i (i)
∈ ≤ ∈
for i = 1,2. If A = A A and β = (β ,β ) then we have that H (β) =
1 2 (1) (2) A
⊕
DH (β )+DH (β ) where H (β ) is a left ideal of the Weyl Algebra D =
A1 (1) A2 (2) A1 (1) A1
C[x ,...,x ] ∂ ,...,∂ and H (β ) is a left ideal of the Weyl Algebra D =
1 n1 h 1 n1i A2 (2) A2
C[x ,...,x ] ∂ ,...,∂ (equivalently, M (β) is the exterior tensor
n1+1 n1+n2 h n1+1 n1+n2i A
product of M (β ) and M (β )).
A1 (1) A2 (2)
The following corollary follows from Lemma 2.2 by general properties of the
exterior tensor product of holonomic D–modules.
3
Corollary 2.3. Under the assumptions of Lemma 2.2 we have:
i) rank(M (β)) = rank(M (β )) rank(M (β )).
A A1 (1) · A2 (2)
ii) If Ω is a basis for the space of (holomorphic)solutions of the hypergeometric
i
system M (β ) at a point p Cni, then the set
Ai (i) i ∈
Ω = f (x ,...,x ) f (x ,...,x ) : f Ω ,i = 1,2
{ 1 1 n1 · 2 n1+1 n1+n2 i ∈ i }
is a basis for the space of (holomorphic) solutions of M (β) at p = (p ,p )
A 1 2
∈
Cn1+n2.
In view of Corollary 2.3, we can already give a first type of families of hyper-
geometric systems for which the rank jump grows exponentially with d.
Theorem 2.4. Let A Zd×n and β Cd be such that M (β) has a rank jump,
A
∈ ∈
i. e. rank(MA(β))/volZA(A) = q > 1. Consider for dr = rd with r 1 the
≥
matrix A Zdr×nr (n = rn), defined as the direct sum of r copies of A, and
r r
∈
the parameter vector β = (β,...,β) Cdr, defined by r copies of β as well. We
r
∈
have that the family given by (Ar,βr) satisfies rank(MAr(βr))/volZAr(Ar) ≥ adr
where a = √d q > 1.
In the sequel, we will first consider an example of a family similar to the ones
given by Theorem 2.4 and then, we will modify this example in order to exhibit
for all d 2 a family of A–hypergeometric systems with exponential growth of
≥
rank jumps on d = rank(A) which are not exterior tensor products of smaller
hypergeometric systems.
Example 2.5. For d = 2 we will consider the first example of a hypergeometric
system with rank jump described in [ST98]. Consider the pair (A ,β ) where
(2) (2)
1 1 1 1 1
A = and β = . (4)
(2) (cid:18) 0 1 3 4 (cid:19) (2) (cid:18) 2 (cid:19)
The toric ideal associated with A is
(2)
I = ∂ ∂ ∂ ∂ ,∂2∂ ∂3,∂ ∂2 ∂3,∂ ∂2 ∂2∂
A(2) 1 4 − 2 3 1 3 − 2 2 4 − 3 1 3 − 2 4
(cid:0) (cid:1)
and the Euler operators are E β = x ∂ + x ∂ + x ∂ + x ∂ 1 and
1 (2),1 1 1 2 2 3 3 4 4
− −
E β = x ∂ +3x ∂ +4x ∂ 2.
2 (2),2 2 2 3 3 4 4
− −
For this example rank(MA (β)) = volZA (A(2)) = 4 for all β C2 β(2)
(2) (2) ∈ \{ }
but rank(M (β )) = 5. A basis of the space of solutions of M (β ) can also
A (2) A (2)
(2) (2)
be found in [ST98]. Let us point out that this basis consists of the two Laurent
polynomials p = x2/x , p = x2/x and other 3 functions that are not Laurent
1 2 1 2 3 4
polynomials.
4
Example 2.6. For d = 3, we will consider the hypergeometric system of the
family M (β ) described in [MW07]. It is the one associated with the
{ A(d) (d) }d≥2
pair
1 1 1 1 1 1 1
A = 0 0 0 0 1 1 and β = 0 (5)
(3) (3)
0 1 3 4 0 1 2
The volume of A is d+2 = 5 while the rank of M (β(3)) is 2d+1 = 7.
(3) A
(3)
Example 2.7. For any d 4, let r,s N be such that 2r + 3s = d. We
≥ ∈
will choose s as high as possible in order to fix uniques r,s N for each d (in
∈
particular 0 r 4).
≤ ≤
We define A Zd×2d to be the direct sum of r copies of the matrix A and
(d) (2)
∈
s copies of the matrix A . By Lemma 2.1 and examples 2.5 and 2.6 we have
(3)
that volZA(d)(A(d)) = 4r5s.
On the other hand, let β Cd be the complex vector with coordinates
(d)
∈
β = 1 and β = 2 for 1 i r and β = 1,β =
(d),2i−1 (d),2i (d),2r+3j−2 (d),2r+3j−1
≤ ≤
0,β = 2 for 1 j s (i. e., β has a copy of β for each copy of A
(d),2r+3j (d) (2) (2)
≤ ≤
and a copy of β for each copy of A ). With this definition of (A ,β ) and us-
(3) (3) (d) (d)
ing Corollary 2.3 and examples 2.5 and 2.6 we have that rank(M (β )) = 5r7s.
A(d) (d)
Thus rank(MA(d)(β(d)))/volZA(d)(A(d)) = (5/4)r(7/5)s ≥ (√5/2)d.
Remark 2.8. Example 2.7 also shows that the rank jump j (β) can be greater
A
than the number of Laurent polynomial solutions of M (β). Indeed, since the
A
space of Laurent polynomial solutions of M (β ) has dimension 2 (see [ST98])
A (2)
(2)
and the space of Laurent polynomial solutions of M (β ) has dimension 4 (see
A (3)
(3)
[MW07]) then, by Corollary 2.3, the space of of Laurent polynomial solutions of
M (β(d)) has dimension 2r4s < j (β ) = 5r7s 4r5s for r,s 1.
A(d) A(d) (d) − ≥
We are going to modify Example 2.7 in order to get hypergeometric systems
that are not exterior tensor products of smaller hypergeometric systems.
Consider the following matrices and parameters:
1 2 2 2 2 3
Aˆ = and βˆ = . (6)
(2) (cid:18) 0 0 1 3 4 (cid:19) (2) (cid:18) 2 (cid:19)
1 2 2 2 2 2 2 3
Aˆ = 0 0 0 0 0 1 1 and parameter βˆ = 0 (7)
(3) (3)
0 0 1 3 4 0 1 2
Notice that Aˆ and Aˆ are obtained from A and A respectively by
(2) (2) (2) (3)
multiplying the first row by 2 (this doesn’t change the hypergeometric system)
andthenbyaddingafirstcolumnwithitsfirstcoordinateequalto1andtheother
coordinates equal to zero. After these modifications we get that vol (Aˆ ) =
ZAˆ (2)
(2)
2 4 = 8 and that vol (Aˆ ) = 2 5 = 10. However, since βˆ is a hole in NAˆ
· ZAˆ(3) (3) · (i) (i)
5
(meaning that βˆ / NAˆ but βˆ +(NAˆ 0 ) NAˆ ) we have by Remark
(i) (i) (i) (i) (i)
∈ \{ } ⊆
4.14 in [Oku06] that rank(M (βˆ )) = vol (Aˆ )+(i 1), i = 2,3.
Aˆ(i) (i) ZAˆ(i) (i) −
The following Lemma follows from the results in [Ber11].
Lemma 2.9. Let A Zd×n and B Zd×m be two matrices verifying that NA =
∈ ∈
NB and ∆ = ∆ then rank(M (β)) = rank(M (β)) for all β Cd.
A B A B
∈
For d = 2r + 3s 2, r,s N (with s as high as possible), let βˆ Cd be
(d)
≥ ∈ ∈
ˆ ˆ
the complex vector that is given by r copies of β and s copies of β . The new
(2) (3)
matrix Aˆ Zd×(6r+8s−1) is constructed as follows.
(d)
∈ r
Let a ,a ,...,a Zd be the columns of the matrix A = Aˆ
1 2 5r+7s r,s (2)
s ∈ ⊕ ···
|{z}
Aˆ Aˆ Aˆ Zd×(5r+7s).
(2) (3) (3)
⊕ ⊕ ··· ⊕ ∈
We will co|n{zst}ruct a matrix Aˆ by adding r + s 1 column vectors to the
(d)
−
matrix A . This vectors will belong to both ∆ and NA . These conditions
r,s Ar,s r,s
guarantee that volZAˆ(d)(Aˆ(d)) = volZAr,s(Ar,s) = 8r10s and by Lemma 2.9, we will
also have that rank(M (β)) = rank(M (β)) for all β Cd. In particular, for
Aˆ(d) Ar,s ∈
β = βˆ , we have rank(M (βˆ )) = 9r12s.
(d) Aˆ (d)
(d)
If r 2 then for 1 i r 1 we define:
≥ ≤ ≤ −
1 1
a = a +a = a + a NA ∆ .
5r+7s+i 1 5i+1 2 2 2 5i+2 ∈ r,s ∩ Ar,s
Notice that (a ) equals 1 for j = 1,2i+1 and 0 otherwise.
5r+7s+i j
If r,s 1 then for 1 i s we define
≥ ≤ ≤
1 1
a = a +a = a + a NA ∆ .
5r+7s+r−1+i 1 5r+7i+1 2 2 2 5r+7i+2 ∈ r,s ∩ Ar,s
If r = 0 and s 1 then for 1 i s 1 we define
≥ ≤ ≤ −
1 1
a = a +a = a + a NA ∆ .
7s+i 1 7i+1 2 2 2 7i+2 ∈ r,s ∩ Ar,s
Let us define Aˆ = (a a ... a ) and recall that βˆ Cd is given by
d 1 2 6r+8s−1 (d)
∈
ˆ ˆ ˆ
r copies of β and s copies of β . The hypergeometric system M (β ) is
(2) (3) Aˆ (d)
(d)
not an exterior tensor product of smaller hypergeometric systems and we have
proved the following.
Theorem 2.10. With the notations above we have
ˆ
rank(M (β ))
Aˆ(d) (d) = (9/8)r(12/10)s ( 9/8)d
vol (Aˆ ) ≥
ZAˆ (d) p
(d)
6
Remark 2.11. Notice that the toric ideal associated with Aˆ is not homoge-
(d)
neous. However, by Theorem 7.3 in [Ber10], if we consider the associated homo-
geneous matrix Aˆh (that is obtained by adding to the matrix Aˆ a first column
(d) (d)
of zeroes and after that a first row of ones) and the parameter βˆh = (β ,βˆ )
(d) 0 (d)
with β C then the rank of M (βˆ ) equals the rank of M (βˆ ) if β C
0 ∈ Aˆh(d) (d) Aˆ(d) (d) 0 ∈
is generic. This implies that for particular β (for example β = 0) the rank of
0 0
M (βˆh ) will be greater than or equal to the rank of M (βˆ ) by the upper
Aˆh(d) (d) Aˆ(d) (d)
semi–continuity of the rank [MMW05]. Moreover, vol (Aˆh ) = vol (Aˆ ).
Aˆh(d) (d) Aˆ(d) (d)
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