Maximally reducible monodromy of bivariate hypergeometric systems
aa r X i v : . [ m a t h . C V ] O c t MAXIMALLY REDUCIBLE MONODROMY OF BIVARIATEHYPERGEOMETRIC SYSTEMS
TIMUR SADYKOV AND SUSUMU TANAB´E
Abstract.
We investigate branching of solutions to holonomic bivariate hypergeometricsystems of Horn’s type. Special attention is paid to the invariant subspace of Puiseuxpolynomial solutions. We mainly study Horn systems defined by simplicial configurationsand Horn systems whose Ore-Sato polygons are either zonotopes or Minkowski sums ofa triangle and segments proportional to its sides. We prove a necessary and sufficientcondition for the monodromy representation to be maximally reducible, that is, for thespace of holomorphic solutions to split into the direct sum of one-dimensional invariantsubspaces. Introduction
To compute the monodromy group of a differential equation or a system of such equa-tions is a notoriously difficult problem in the analytic theory of differential equations.One of the reasons for this is that the computation of the monodromy group requires fullunderstanding of the structure of the solution space of the system of differential equa-tions under study, including the dimension of this space, a basis in it, the fundamentalgroup of the complement to singularities of the system as well as analytic continuationand branching properties of the chosen basis.The purpose of the present paper is to investigate the monodromy of certain families ofsystems of partial differential equations of hypergeometric type. It uses and extends theresults in [17] and [18]. While the monodromy group of the classical Gauss second-orderhypergeometric differential equation has been computed by Schwarz and the monodromyof the ordinary generalised hypergeometric equation has been described in [3], the problemof finding the monodromy group of a general hypergeometric system of partial differentialequations remains unsolved despite all the effort and several well-understood special cases(see [1], [2] and the references therein). The original motivation for the results presentedin the paper goes back to the work [4] where the authors have posed the problem ofdescribing the Gelfand-Kapranov-Zelevinsky (GKZ) nonconfluent hypergeometric systems(see [9]), whose solution space contains a nonzero rational function for a suitable choiceof its parameters. In terms of monodromy, this is equivalent to the existence of a one-dimensional subspace in the space of holomorphic solutions to the system under studywith the trivial action of monodromy on it.
The first author was supported by the grant of the Government of the Russian Federation for in-vestigations under the guidance of the leading scientists of the Siberian Federal University (contractNo. 14.Y26.31.0006), by the grants of the Russian Foundation for Basic Research (grants no. 13-01-12417-ofi-m2, 15-31-20008-mol-a-ved), as well as by the Japanese Society for the Promotion of Science.The second author was supported by JSPS grant no. 20540086.
In the present paper, we solve a closely related problem of describing all holonomicbivariate hypergeometric systems in the sense of Horn (see [5] and the references therein)whose solution space splits into a direct sum of one-dimensional monodromy invariantsubspaces (Theorem 6.1). We call such a monodromy representation maximally reducible.
The relation between GKZ and Horn hypergeometric systems has been studied in detail inSection 5 of [5]: for any GKZ system there exists a canonically defined Horn system and anaturally defined bijective map from a subspace in the space of its analytic solutions intothe space of solutions to the GKZ system. The solutions of the Horn system that are nottaken into account by this map are its persistent Puiseux polynomial solutions in the senseof Definition 2.10 below. Here and throughout the paper by a Puiseux polynomial we meana finite linear combination of monomials with (in general) arbitrary complex exponents.As it has been announced in Theorem 5.3 of [5], persistent polynomial solutions are thecokernel of the map from GKZ solutions to Horn system solutions.In our formulation, the above mentioned question of [4] can be answered in the followingmanner. The dimension of the space of non-persistent Puiseux polynomial solutions to aHorn system is equal to that of the space of Puiseux polynomial solutions to the corre-sponding GKZ system. For the bivariate Horn system, full characterisation of persistentsolutions is given in Proposition 2.12 and Corollary 4.2.The authors are thankful to the referee for the careful reading of the manuscript andnumerous suggestions that have led to a substantial improvement of the paper. Publica-tion of the paper in the present special issue of the journal is a tribute to A. A. Bolibrukhfor his constant support of the second author (S.T.) over many years.2.
Notation, definitions and preliminaries
Throughout the paper, the following notation will be used: n = the number of x variables; m = the number of rows in the matrix defining the Horn system; ν ( a , b ; a , b ) ≡ ν (cid:18) a b a b (cid:19) = the index of the two vectors ( a , b ) , ( a , b ) , see Def-inition 2.6;for m = ( m , . . . , m n ) , | m | = P ni =1 m i and m ! = m ! . . . m n !;for x = ( x , . . . , x n ) and m = ( m , . . . , m n ) , x m = x m . . . x m n n ; Z ≥ = the set of non-negative integers, Z ≤ = the set of non-positive integers;Horn( ϕ ) = the Horn hypergeometric system defined by the Ore-Sato coefficient ϕ, seeDefinition 2.3.Horn( A, c ) = the Horn hypergeometric system defined by the Ore-Sato coefficient (2.2)with t i = 1 for any i = 1 , . . . , n and U ( s ) ≡ . See the construction after Definition 2.3;Ψ( ϕ ) = the subspace of Puiseux polynomial solutions to the Horn system defined bythe Ore-Sato coefficient ϕ, see Definition 2.3;Ψ ( ϕ ) ⊂ Ψ( ϕ ) is the subspace of persistent Puiseux polynomial solutions to the Hornsystem defined by the Ore-Sato coefficient ϕ , see Definition 2.10; F = the set of all pure fully supported solutions to a Horn system. Observe that it isin general not a linear subspace since the intersection of the domains of convergence ofall elements in F may be empty; AXIMALLY REDUCIBLE MONODROMY OF BIVARIATE HYPERGEOMETRIC SYSTEMS 3 F x (0) = the linear space of fully supported solutions to a Horn system which convergeat a nonsingular point x (0) ; A ( ϕ ) = the amoeba of the singularity of an Ore-Sato coefficient ϕ ; see Definition 5.1; C ∨ = the dual of a convex cone C ; P ( ϕ ) is the polygon of the Ore-Sato coefficient ϕ, see Definition 2.5.For an Ore-Sato coefficient ϕ and ζ ∈ R n we set M ( ϕ, ζ ) = (cid:26) the connected component of c A ( ϕ ) which contains ζ , if ζ ∈ c A ( ϕ ) , R n , if ζ ∈ A ( ϕ ); S (Horn( A, c )) is the space of solutions to the system Horn(
A, c ), that are holomorphicaway from the singular hypersurface.
Definition 2.1.
A formal Laurent series(2.1) X s ∈ Z n ϕ ( s ) x s is called hypergeometric if for any j = 1 , . . . , n the quotient ϕ ( s + e j ) /ϕ ( s ) is a rationalfunction in s = ( s , . . . , s n ) . Throughout the paper we denote this rational function by P j ( s ) /Q j ( s + e j ) . Here { e j } nj =1 is the standard basis of the lattice Z n . By the support of this series we mean the subset of Z n on which ϕ ( s ) = 0 . We say that such a seriesis fully supported, if the convex hull of its support contains (a translation of) an open n -dimensional cone.A hypergeometric function is a (multi-valued) analytic function obtained by means ofanalytic continuation of a hypergeometric series with a nonempty domain of convergencealong all possible paths. Theorem 2.2. (Ore, Sato [8], [19])
The coefficients of a hypergeometric series are givenby the formula (2.2) ϕ ( s ) = t s U ( s ) m Y i =1 Γ( h A i , s i + c i ) , where t s = t s . . . t s n n , t i , c i ∈ C , A i = ( A i, , . . . A i,n ) ∈ Z n , i = 1 , . . . , m, and U ( s ) is aproduct of certain rational function and a periodic function φ ( s ) s.t. φ ( s + e j ) = φ ( s ) forevery j = 1 , . . . , n . In the article [19] Appendix (A.3) a precise description of rational function factor of U ( s ) is available.We will call any function of the form (2.2) the Ore-Sato coefficient of a hypergeometricseries. We remark that in view of the formulasin( πz )Γ(1 − z )Γ( z ) = π an Ore-Sato coefficient can be a function of the form ϕ ( s ) = t s Y i ∈ I Γ( h A i , s i + c i ) Y j / ∈ I e π √− h A j ,s i + c j ) Γ(1 − h A j , s i − c j ) , where I ⊂ { , . . . , m } . TIMUR SADYKOV AND SUSUMU TANAB´E
Given the above data ( t i , c i , A i , U ( s )) that determines the coefficient of a hypergeomet-ric series, it is straightforward to compute the rational functions P i ( s ) /Q i ( s + e i ) usingthe Γ-function identity. The converse requires solving a system of difference equationswhich is only solvable under some compatibility conditions on P i , Q i . A careful analysisof this system of difference equations has been performed in [14].In this paper the Ore-Sato coefficient (2.2) plays the role of a primary object whichgenerates everything else: the series, the system of differential equations, the algebraichypersurface containing the singularities of its solutions, the amoeba of its defining polyno-mial, and, ultimately, the monodromy group of the hypergeometric system of differentialequations. We will also assume that m ≥ n since otherwise the corresponding hypergeo-metric series (2.1) is just a linear combination of hypergeometric series in fewer variables(times arbitrary function in remaining variables that makes the system non-holonomic)and n can be reduced to meet the inequality. Definition 2.3.
The Horn system of an Ore-Sato coefficient.
A (formal) Laurent series P s ∈ Z n ϕ ( s ) x s whose coefficient satisfies the relations ϕ ( s + e j ) /ϕ ( s ) = P j ( s ) /Q j ( s + e j ) is a(formal) solution to the following system of partial differential equations of hypergeometrictype(2.3) x j P j ( θ ) f ( x ) = Q j ( θ ) f ( x ) , j = 1 , . . . , n. Here θ = ( θ , . . . , θ n ) , θ j = x j ∂∂x j . The system (2.3) will be referred to as the Horn hyperge-ometric system defined by the Ore-Sato coefficient ϕ ( s ) (see [8]) and denoted by Horn( ϕ ).We shall denote by S (Horn( ϕ )) the solution space to Horn( ϕ ). In this paper we treat onlyholonomic Horn hypergeometric systems if not otherwise specified i.e. rank(Horn( ϕ )) isalways assumed to be finite. A necessary and sufficient condition for a system Horn( ϕ )to be holonomic has been established in [6], Theorem 6.3.We will often be dealing with the important special case of an Ore-Sato coefficient (2.2)where t i = 1 for any i = 1 , . . . , n and U ( s ) ≡ . The Horn system associated with such anOre-Sato coefficient will be denoted by Horn(
A, c ) , where A is the matrix with the rows A , . . . , A m ∈ Z n and c = ( c , . . . , c m ) ∈ C m . In this case the following operators P j ( θ )and Q j ( θ ) explicitly determine the system (2.3): P j ( s ) = Y i : A i,j > A i,j − Y ℓ ( i ) j =0 (cid:16) h A i , s i + c i + ℓ ( i ) j (cid:17) ,Q j ( s ) = Y i : A i,j < | A i,j |− Y ℓ ( i ) j =0 (cid:16) h A i , s i + c i + ℓ ( i ) j (cid:17) . Definition 2.4.
The Ore-Sato coefficient (2.2), the corresponding hypergeometric se-ries (2.1), and the associated hypergeometric system (2.3) are called nonconfluent if(2.4) m X i =1 A i = 0 . AXIMALLY REDUCIBLE MONODROMY OF BIVARIATE HYPERGEOMETRIC SYSTEMS 5
It is a well known fact (e.g. [6], Theorem 6.3) that a nonconfluent holonomic hypergeo-metric system is a regular holonomic system i.e. every solution admits polynomial growthwhen approaching its singular loci.
Definition 2.5.
The polygon of a nonconfluent Ore-Sato coefficient in two variables.
Using, if necessary, the Gauss multiplication formula for the Γ-function and N ∈ N ,Γ( h A i , s i + c i ) = N h A i ,s i + c i (2 π ) ( N − / √ N Γ (cid:18) h A i , s i + c i N (cid:19) Γ (cid:18) h A i , s i + c i + 1 N (cid:19) . . . Γ (cid:18) h A i , s i + c i + N − N (cid:19) , we may without loss of generality assume that for any i = 1 , . . . , p the nonzero componentsof the vector A i are relatively prime. Let l i denote the generator of the sublattice { s ∈ Z : h A i , s i = 0 } and let k i be the number of elements in the multiset { A , . . . , A m } whichcoincide with A i . The nonconfluency condition (2.4) implies that there exists a uniquelydetermined (up to a translation) integer convex polygon whose sides are translations ofthe vectors k i l i , the vectors A , . . . , A m being the outer normals to its sides. The numberof sides of this polygon coincides with the number of different elements in the multiset ofvectors { A , . . . , A m } . We call this polygon the polygon of the Ore-Sato coefficient (2.2) and denote it by P ( ϕ ) . Conversely, any convex integer polygon determines a ( m × A ( P ) , c ). Thisrelation is illustrated by example 4.5. Definition 2.6.
For a pair of vectors ( a , b ) , ( a , b ) ∈ Z we set ν ( a , b ; a , b ) = min( | a b | , | b a | ) , if ( a , b ), ( a , b ) arein opposite open quadrants of Z , , otherwise . The number ν ( a , b ; a , b ) is called the index associated with the lattice vectors ( a , b )and ( a , b ). The index of the rows of a 2 × M will be denoted by ν ( M ) . Definition 2.7.
By the initial exponent of a multiple hypergeometric series x α X s ∈ Z n ϕ ( s ) x s we mean the vector α = ( α , . . . , α n ) ∈ C n . Observe that the initial exponent of such aseries is only defined up to shifts by integer vectors. However, in the view of Proposi-tion 3.11 and Corollary 3.13 (to be proved in Section 3) this is exactly what we need forcomputing monodromy of hypergeometric systems.
Definition 2.8.
The support of a series solution to (2.3) is called irreducible if thereexists no series solution to (2.3) supported in its proper nonempty subset.
Definition 2.9.
A series solution with irreducible support f ( x ) = P α ∈ Λ c α x α to a Hornsystem is called pure if for any α, β ∈ Λ we have α = β mod Z n . In other words, a series(in particular, a polynomial) solution centered at the origin and with irreducible support
TIMUR SADYKOV AND SUSUMU TANAB´E is called pure if it is given by the product of a monomial and a Laurent series. A setof linearly independent series { f k ( x ) } rk =1 is called a pure basis of the solution space ofa Horn system in a neighborhood of a nonsingular point x ∈ C n if every f k convergesat x, is a pure solution and together they span a linear space whose dimension equals theholonomic rank of the Horn system.Since a Horn system has polynomial coefficients, it follows that any of the Puiseux seriessolutions to a holonomic Horn system can be written as a finite linear combination of puresolutions to the same system of equations. Here the holonomic property is necessary toensure that the linear combination is finite. Moreover, in a neighborhood of a nonsingularpoint, a pure basis in the local solution space of a Horn system is defined uniquely up topermutation and multiplication of its elements with nonzero constants. In this paper wewill neglect this unessential difference between pure bases of solutions to hypergeometricsystems. If necessary, we will explicitly specify the ordering of the elements of the purebasis and the way they are normalized. The pure basis of a hypergeometric system isespecially convenient for computing monodromy since, within the domain of convergenceof the basis series, the monodromy matrices are diagonal. Definition 2.10.
A Puiseux polynomial solution to the hypergeometric system Horn(
A, c )is called persistent if its support remains finite under arbitrary small perturbations of thevector of parameters c. For instance, the first solution to the hypergeometric system (3.5) is a persistent Puiseuxmonomial since it remains monomial for any ( c , c , c ) ∈ C . The second solution to (3.5)is a (Puiseux) polynomial only for − ( c + c + c ) ∈ N and it is therefore not a persistentpolynomial solution. The notion is also illustrated in Examples 4.5, 6.8 and 6.9.We will denote the linear space of all (not necessarily persistent) Puiseux polynomialsolutions to the Horn system defined by the Ore-Sato coefficient ϕ ( s ) by Ψ( ϕ ) and use thenotation Ψ ( ϕ ) for the space of all persistent polynomial solutions to this system. Thefollowing is an immediate consequence of Definition 2.10. Proposition 2.11.
For an Ore-Sato coefficient ϕ defined by (2.2) with generic vector c =( c , . . . , c m ) ∈ C m of parameters every Puiseux polynomial solution to the correspondinghypergeometric system Horn ( ϕ ) is persistent. That is to say, Ψ( ϕ ) = Ψ ( ϕ ) as long as c is generic. The next proposition is proved by analysis of the difference equations satisfied by thecoefficient of a hypergeometric polynomial (see [5]).
Proposition 2.12.
Let ϕ ( s ) be an Ore-Sato coefficient and let f ( x ) be a Puiseux poly-nomial solution to Horn( ϕ ) . If this polynomial solution is persistent then there exists amulti-index I = { i , . . . , i n } ⊂ { , . . . , m } with different components such that for any s ∈ supp f and any ℓ = 1 , . . . , n there exists j ∈ I and k ∈ { , . . . , | A j,ℓ | − } such that h A j , s i + c j + k = 0 . Definition 2.13.
We say that the Ore-Sato coefficient ϕ ( s ) = Q mi =1 Γ( h A i , s i + c i ) (aswell as the corresponding hypergeometric system Horn( ϕ ( A, c )) is resonant if there existsa multi-index I = ( i , . . . , i k ) with 1 ≤ i < . . . < i k ≤ m, ≤ k ≤ m such that for AXIMALLY REDUCIBLE MONODROMY OF BIVARIATE HYPERGEOMETRIC SYSTEMS 7 any linear relation a i A i + . . . + a i k A i k = 0 with integer and relatively prime coefficients a i , . . . , a i k ∈ Z we have a i c i + . . . + a i k c i k ∈ Z . The system Horn( ϕ ( A, c )) is called maximally resonant if the above holds for any multi-index I = ( i , . . . , i k ) such that thecorresponding integer vectors A i , . . . , A i k are linearly dependent.The notion of resonance is illustrated by the following example that is based on ahypergeometric system of the smallest possible rank. Example 2.14.
To simplify the notation, here and throughout the paper we will definea system of linear homogeneous differential equations by giving the set of its generatingoperators. The Horn system(2.5) (cid:26) x ( θ + θ + c ) − ( θ + c ) ,x ( θ + θ + c ) − ( θ + c )is the only (up to a monomial change of variables defined by a unimodular matrix) bivari-ate hypergeometric system whose holonomic rank equals 1 for all values of its parameters c , c , c ∈ C . The only solution to this system is x − c x − c (1 − x − x ) c + c − c . It is resonant(and maximally resonant as well, since it has holonomic rank 1) if and only if c + c − c ∈ Z . The monodromy of (2.5) only depends on the values of a, b, c modulo Z and is the sub-group of C with the three generators { exp(2 π √− c ) , exp(2 π √− c ) , exp(2 π √− c ) } in non-resonant case, while it has less than two generators in resonant case (if the groupis not trivial).The crucial importance of the notion of resonance will be revealed in the theorems andexamples that follow. Roughly speaking, nonresonant parameters of a hypergeometricsystem mean that any of its solutions is either a fully supported series (centered at theorigin) or a persistent Puiseux polynomial. Resonant parameters may correspond to non-holonomic systems, systems with non-persistent polynomial solutions, non-fully supportedseries solutions or, possibly, logarithmic solutions which do not admit any expansions intoPuiseux series (centered at the origin) at all. For instance, the hypergeometric system (2.6)is maximally resonant. Definition 2.15.
A solution f ( x ) to the system of differential equations Horn( ϕ ) at anonsingular point x (0) ∈ C n is said to generate a linear subspace L ⊂ S (Horn( ϕ )) | V ( x (0) ) ofthe space of all holomorphic solutions to Horn( ϕ ) in a simply connected neighbourhood V ( x (0) ) if every element of L can be represented as a linear combination of branchesof f ( x ) on V ( x (0) ) . We will say that f ( x ) is a generating solution of L. A function iscalled a generating solution to a system of equations if it generates the whole space of itsholomorphic solutions at any nonsingular point. In Section 4 we will construct generatingsolutions for two families of hypergeometric systems (Proposition 4.4, Proposition 4.7).
Example 2.16.
The maximally resonant Horn system defined by the Ore-Sato coefficient ϕ ( s ) = Γ( s )Γ( s )Γ( s + s )Γ( − s ) Γ( − s ) is given by(2.6) (cid:26) x θ ( θ + θ ) − θ ,x θ ( θ + θ ) − θ . This system has holonomic rank 4. Its space of holomorphic solutions is spanned by1 , log x , log x , log x log x + PolyLog(2 , x ) +PolyLog(2 , x ) . Here PolyLog (2 , z ) =
TIMUR SADYKOV AND SUSUMU TANAB´E P ∞ k =1 z k /k . The resultant of the principal symbols of (2.6) equals x x ( x − x − x + x − . Using the properties of PolyLog(2 , z ) (see [10]), we conclude that themonodromy group of (2.6) is generated by the four matrices M x =0 = π √− π √−
10 0 1 00 0 0 1 , M x =0 = π √− π √−
10 0 0 1 ,M x =1 = − π √− , M x =1 = − π √− . This monodromy representation shows that log x log x + PolyLog(2 , x ) + PolyLog(2 , x )is a generating solution of S (Horn( ϕ )) . If the monodromy representation of the entire solution space S (Horn( ϕ )) is irreduciblethen it admits a generating solution. On the other hand, the monodromy representationcan be reducible for S (Horn( ϕ )) with a generating function as the above Example 2.16illustrates.The main result in the paper (Theorem 6.1) describes bivariate hypergeometric systemswhose solution spaces split into one-dimensional invariant subspaces. Throughout thepaper, we will adopt the following definition. Definition 2.17.
We will say that the monodromy representation of a system of equationsis maximally reducible if its solution space splits into a direct sum of one-dimensionalinvariant subspaces.3.
The structure of the space of holomorphic solutions to a Hornsystem
Integral representations and calculation of multidimensional residues.
Ourmain tool for computing analytic continuation of a hypergeometric series is the Mellin-Barnes integral. The following theorem gives an integral representation for solutions to ahypergeometric system.
Theorem 3.1. (See [14]).
Let ψ ( s ) = m Y j =1 Γ( h A j , s i + c j ) be a nonconfluent Ore-Sato coefficient. Let us put ϕ ( s ) = φ ( s ) ψ ( s ) , where φ ( s ) is aperiodic meromorphic function with the period 1 in every coordinate direction. Then theMellin-Barnes integral (3.1) M B ( ϕ, C ) := Z C ϕ ( s ) x s ds AXIMALLY REDUCIBLE MONODROMY OF BIVARIATE HYPERGEOMETRIC SYSTEMS 9 represents a solution to
Horn(
A, c ) . Here C is any n -dimensional contour which is homol-ogous to its unitary shifts in any real direction in the complement of the singularities ofthe integrand in (3.1). The next proposition is proved, like the previous theorem, by computing multidimen-sional residues at simple singularities. It allows one to convert a multiple hypergeometricseries into an iterated Mellin-Barnes integral.
Proposition 3.2.
Let ψ ( k ) /k ! be a nonconfluent Ore-Sato coefficient with generic param-eters, A ∈ GL ( n, Z ) an integer nondegenerate square matrix with the rows A , . . . , A n and α ∈ C n . For a sufficiently small ε > and k ∈ N n let τ ( k ) = { s ∈ C n : |h A j , s i + α j + k j | = ε, for any j = 1 , . . . , n } and define C = P k ∈ N n τ ( k ) . Then X k ∈ N n ( − | k | k ! ψ ( k ) x Ak + α = 1(2 π √− n | A | Z C n Y j =1 Γ(( − A − ( s − α )) j ) ψ ( A − ( s − α )) x s ds. The following theorem gives a solution to the hypergeometric system Horn(
A, α ) in theform of a multiple Mellin-Barnes integral and allows one to convert it into a hypergeomet-ric (Puiseux) series by computing the residues at a distinguished family of singularities ofthe integrand.
Theorem 3.3. (See [14]).
Let A be a m × n integer matrix of full rank n with therows A , . . . , A m and let I = ( i , . . . , i n ) ⊂ { , . . . , m } be a multi-index such that thematrix A I with the rows A i , . . . , A i n is nondegenerate. For a sufficiently small ε > and k ∈ N n let τ I ( k ) = { s ∈ C n : |h A i j , s i + α i j + k j | = ε, for any j = 1 , . . . , n } anddefine C I = P k ∈ N n τ I ( k ) . Then for generic α ∈ C m and α I = ( α i , . . . , α i n ) the followingMellin-Barnes integral satisfies the system of equations Horn(
A, α ) and can be representedin the form of a hypergeometric (Puiseux) series: (3.2) 1(2 π √− n Z C I m Y j =1 Γ( h A j , s i + α j ) x s ds = X k ∈ N n ( − | k | k ! | A I | Y j I Γ( h A j , − A − I ( k + α I ) i + α j ) x − A − I ( k + α I ) . Holonomic rank formulas.
To give a proper formulation to the main Theorem 3.7of this section, we introduce the following notion.
Definition 3.4.
For m ≥ n let A be a m × n integer matrix of rank n with the rows A , . . . , A m and let c ∈ C m be a vector of parameters. Let I = ( i , . . . , i n ) be a multi-index such that the square matrix A I with the rows A i , . . . , A i n is nondegenerate. Let c I denote the vector ( c i , . . . , c i n ) . The hypergeometric system Horn( A I , c I ) will be referredto as an atomic system associated with the system Horn(
A, c ) . The number of atomic sys-tems associated with a hypergeometric system Horn(
A, c ) equals the number of maximalnondegenerate square submatrices of the matrix A. It follows from Theorem 1.3 in [15] that, as long as the supports of series solutions areconcerned, a generic hypergeometric system is built of associated atomic systems. Moreprecisely, the set of supports of solutions to a hypergeometric system with generic pa-rameters consists of supports of solutions to associated atomic systems. In particular, theinitial exponents of Puiseux polynomial solutions to a hypergeometric system are preciselythe initial exponents of Puiseux polynomials which satisfy the associated atomic systems.In the following statement we sum up the basic properties of Horn hypergeometric systemsthat we will need in the sequel.
Proposition 3.5.
For any solution v ( x ) to an atomic system associated with a noncon-fluent holonomic system Horn(
A, c ) with a generic vector of parameters c ∈ C m , thereexists a solution u ( x ) ∈ S (Horn( A, c )) whose support coincides with the support of thefunction v ( x ) .Proof. Consider a nonconfluent holonomic system Horn(
A, c ) defined by the Ore-Satocoefficient ϕ ( s ) = φ ( s ) m Y i =1 Γ( h A i , s i + c i )with a suitable meromorphic periodic function φ ( s ).Any solution to the associated atomic system Horn( A I , c I ), I = ( i , . . . , i n ) ⊂ { , . . . , m } admits the integral representation v ( x ) = Z C I Y i ∈ I Γ( h A i , s i + c i ) φ ( s ) x s ds for a suitable choice of the contour C I and the periodic function ψ ( s ).Using this integral representation we obtain the following solution to the nonconfluentholonomic system Horn( A, c ): u ( x ) = Z C I Y i ∈ I Γ( h A i , s i + c i ) Y j / ∈ I Γ( h A j , s i + c j ) φ ( s ) x s ds. Since the vector of parameters c ∈ C m is generic, we may assume that the contour C I only contains intersections of n polar sets of the product Q i ∈ I Γ( h A i , s i + c i ), that aremoreover disjoint from the poles of the product Q j I Γ( h A j , s i + c j ) φ ( s ). Thus in a smallneighborhood of the poles of the factor Q i ∈ I Γ( h A i , s i + c i ) the meromorphic function Q j / ∈ I Γ( h A j , s i + c j ) φ ( s ) is holomorphic. This immediately yields that the support of u ( x )coincides with the support of v ( x ). (cid:3) Remark 3.6.
If the vector of parameters c ∈ C m is not generic then the support of asolution u ( x ) ∈ S (Horn( A, c )) to a hypergeometric system can be a proper subset of thesupport to a solution v ( x ) ∈ S (Horn( A I , c I )) of the associated atomic system.Consider the following example: A = (( − , , (2 , − , ( − , − , c = (0 , , − . Given a solution to the hypergeometric system Horn(
A, c ) w ( x ) = X m,n ≥ Res − s +2 s = − m s − s = − n Γ( − s + 2 s )Γ( − s − s − s − s ) x s , AXIMALLY REDUCIBLE MONODROMY OF BIVARIATE HYPERGEOMETRIC SYSTEMS 11 we define the solution to the associated atomic system v ( x ) = X m,n ≥ Res − s +2 s = − m s − s = − n Γ( − s + 2 s )Γ(2 s − s ) x s . Since the solution space S (Horn( A, c )) is invariant under the monodromy action, thefunction u ( x ) = 12 π √− (cid:0) w ( x e π √− , x ) − w ( x , x ) (cid:1) satisfies the system Horn( A, c ). Straightforward computation shows that u ( x ) = (cid:0) x / x / + √ x + √ x (cid:1) x / x / , i.e. the support of u ( x ) consists of the six points { s ∈ C : s − s ∈ Z ≥ , − s + s ∈ Z ≥ , − ≤ s + s ≤ } . Observe that the meromorphic function Γ( − s + 2 s )Γ( − s − s − s − s ) x s has triple poles at the point that belong to the support of u ( x ), allthe other its poles being simple.The next theorem summarizes the main properties of the space of holomorphic solutionsto a Horn system that we need throughout the rest of the paper. Theorem 3.7.
Assume that the hypergeometric system
Horn(
A, c ) is nonconfluent, holo-nomic and has generic vector of parameters c . (1) The space of local holomorphic solutions at a nonsingular point x (0) to Horn(
A, c ) admits the following decomposition: S (Horn( A, c )) = Ψ ⊕ F x (0) . Here Ψ is the subspace ofits persistent Puiseux polynomial solutions and F x (0) is the subspace of its fully supportedPuiseux series solutions which converge at x (0) . (2) The dimension of the space F x (0) of Puiseux series (centered at the origin) whichsatisfy Horn(
A, c ) and converge at x (0) ∈ c A ( ϕ ( A, c )) is given by dim C F x (0) = X I = ( i , . . . , i n ) ⊂ { , . . . , m } M ( ϕ ( A, c ) , Log x (0) ) ⊂ ( A − I R n + ) ∨ | det A I | . (3) The dimension of the space Ψ of persistent Puiseux polynomial solutions to abivariate system Horn(
A, c ) is given by dim C Ψ = P A i , A j lin . indep . ν ( A i , A j ) . Proof. (1) Observe that any Puiseux series solution (centered at the origin) of a Hornsystem with generic parameters is either a fully supported series or a persistent Puiseuxpolynomial. Indeed, for a polynomial to be a solution to a hypergeometric system, itsexponents must satisfy a system of linear algebraic equations. The generic parametersassumption implies that the right-hand-sides of these equations are also generic and hencethe system of linear algebraic equations is defined by a square nondegenerate matrix. Thecorresponding solutions to the hypergeometric system are precisely persistent polynomials.This means, in particular, that for an Ore-Sato coefficient ϕ with generic parametersΨ( ϕ ) = Ψ ( ϕ ) . Since no linear combination of elements in Ψ( ϕ ) can yield a fully supportedPuiseux series, it follows that the sum is direct. (2) This follows from the previous part together with the two-sided Abel lemma (seeLemma 11 in [12]) which describes the domain of convergence of a nonconfluent hypergeo-metric series. By the first part of the theorem the generic parameters assumption impliesthat only fully supported series must be taken into account and it is therefore sufficientto consider square nondegenerate submatrices of A .(3) This is the statement of Theorem 6.6 in [5]. (cid:3) The following result (see [5]) gives the holonomic rank of a bivariate nonconfluent Hornsystem with generic parameters.
Theorem 3.8. ([5])
Let A be an m × integer matrix of full rank such that its rows A , . . . , A m satisfy A + . . . + A m = 0 . If c ∈ C m is a generic parameter vector, then theideal Horn(
A, c ) is holonomic. Moreover, rank(Horn( A, c )) = X i : A i, > A i, · X i : A i, > A i, − X A i , A j lin. dep. ν ( A i , A j ) , where the summation runs over linearly dependent pairs A i , A j of rows of A that lie inopposite open quadrants of Z . Remark 3.9.
The conclusion of Theorem 3.8 only holds under the nonconfluency as-sumption on the matrix A. For instance, the confluent Horn system generated by theoperators x ( θ + θ )( θ + θ − a ) − θ and x ( θ + θ )( θ + θ − a ) − θ is holonomic withrank 2. Indeed, if the above equations are satisfied by a function f ( x ) then f x = f x andhence f ( x ) = g ( x + x ) for a suitable univariate function g. Moreover g ( t ) is a solution tothe ordinary differential equation t g ′′ ( t )+((1 − a ) t − g ′ ( t ) = 0 . A fundamental system ofsolutions of this equation is 1 , Γ( − a, /t ) , where Γ( p, q ) is the incomplete gamma-function.Thus a basis in the solution space of the Horn system is 1 , Γ (cid:16) − a, x + x (cid:17) . Observe thatΓ(1 , / ( x + x )) = e − / ( x + x ) . Thus for a confluent system the rank can be smaller thanthe product of the degrees of the operators even if no parallel lines or persistent polynomialsolutions are present.
Remark 3.10.
Theorem 3.8 is substantially bivariate, yet it can be generalised to arbi-trary dimension of the space of variables. Theorem 6.10, 7.13 in [6] provide an explicitcombinatorial formula for the holonomic rank of a nonconfluent hypergeometric systemHorn(
A, c ) . Let us choose a ( m − n ) × m submatrix B of the matrix A with integer co-efficients whose columns span Z m − n as a lattice, satisfying B · A = 0 ∈ Z m − n × Z n . For g = | ker( B ) / Z A | the index of the integer lattice generated by the columns of A in itssaturation, the following formula holds for generic c ∈ C m :rank(Horn( A, c )) = g vol( B ) + rank(Ψ ( ϕ )) , where vol( B ) denotes the normalised volume of the convex hull of the columns of B . Thisformula is a numerical counterpart of the decomposition Theorem 3.7, 1) on the space ofholomorphic solutions to a hypergeometric system.In example 3.14 we will see that rank(Ψ ) = 1, as Ψ is generated by f and the rankrank ( Horn ( A, ( c , c , c ))) = 2. In fact, for − ( c + c + c ) / ∈ N the rank of fully supportedsolutions is 1 while for − ( c + c + c ) ∈ N the rank of the factor space Ψ / Ψ is 1. AXIMALLY REDUCIBLE MONODROMY OF BIVARIATE HYPERGEOMETRIC SYSTEMS 13
Monodromy action on the invariant subspace of Puiseux polynomial so-lutions.
Recall that by a Puiseux polynomial we mean a finite linear combination ofmonomials with (in general) arbitrary complex exponents. Such a polynomial may onlyhave singularities on the union of the coordinate hyperplanes { x ∈ C n : x . . . x n = 0 } . The set of all Puiseux polynomial solutions of a Horn system is a linear subspace Ψ inthe space of its local holomorphic solutions. This subspace is clearly invariant under theaction of monodromy.Let { p k ( x ) } pk =1 be a pure basis of the linear space Ψ (see Definition 2.9). That is, let p k ( x ) = x v k ˜ p k ( x ) , where v k ∈ C n and ˜ p k ( x ) is a Laurent polynomial (i.e., a polynomialwith integer exponents). Since a Laurent polynomial has no branching, it follows thatthe branching of this basis is the same as that of a system of monomials x v , . . . , x v p , where v k ∈ C n . Thus the branching locus for the solutions of such a Horn system is { x ∈ C n : x . . . x n = 0 } , the generators of the fundamental group with the base point(1 , . . . ,
1) are γ j = (1 , . . . , , e π √− t , , . . . , , t ∈ [0 , , j = 1 , . . . , n. The correspondingmonodromy matrix is given by M j = diag( e π √− v j ) . Intertwining operators for Horn systems.
The purpose of this subsection is tocompute the intertwining operators for the monodromy representations of Horn systemswhose parameters differ by integers. This will allow us to conclude that certain mon-odromy representations are equivalent. The intertwining operators for the monodromyrepresentations of an ordinary hypergeometric differential equation have been computedin [3].Recall that by S (Horn( A, α )) we denote the linear space of (local) solutions to thehypergeometric system Horn(
A, c ) . The class of hypergeometric functions is closed undermultiplication with Puiseux monomials. More precisely, the operator x λ • which multiplesa function with the monomial x λ = x λ . . . x λ n n is a vector space isomorphism between thefollowing spaces: x λ • : S (Horn( A, Aλ + α )) → S (Horn( A, α )) . Since multiplication with a Laurent monomial does not alter the branching of a function,we conclude that for λ ∈ Z n the hypergeometric systems Horn( A, α ) and Horn(
A, Aλ + α )have the same monodromy. Proposition 3.11.
Let A , . . . , A m ∈ Z n be the rows of an integer matrix A of full rank n and let c ∈ C m be the vector of parameters. The differential operator (3.3) h A j , θ i + c j − S (Horn( A, c − e j )) → S (Horn( A, c )) is an intertwining operator for the monodromy representations of the corresponding Hornsystems.Proof. Denote by H i ( A, c ) the differential operator defining the i -th equation in the hy-pergeometric system Horn( A, c ) , (2.3).The following equalities immediately yield the statement: for A i,j ≤ h A j , θ − e i i + c j − H i ( A, c − e j ) = H i ( A, c )( h A j , θ i + c j − , while for A i,j > h A j , θ i + c j − H i ( A, c − e j ) = H i ( A, c )( h A j , θ i + c j − . (cid:3) By means of the intertwining operators, we establish a statement analogous to Propo-sition 2.7 in [3].
Proposition 3.12.
Suppose that the solution space of the system S (Horn( A, c + ℓ )) con-tains a nontrivial subspace of persistent Puiseux polynomial solutions Ψ = { } for ℓ ∈ Z n . Then there is a non-trivial monodromy invariant subspace of S (Horn( A, c )) with codimen-sion higher than . In particular monodromy representation of S (Horn( A, c )) is reducible.Proof. Let J be the set of indices J ⊂ { , . . . , m } such that ker( h A j , θ i + c j + ℓ j ) ∩ Ψ ∋ x α = 0 for j ∈ J. We remark here that we can always find a monomial element in Ψ aslong as Ψ = { } . Then( h A j , θ i + c j + ℓ j ) : S (Horn( A, c + ℓ )) → S (Horn( A, c + ℓ + e j ))has a non-trivial kernel. Assume ℓ j < k j , ℓ j ≤ k j ≤ − h A j , θ i + c j + k j : S (Horn( A, c + ℓ + ( k j − ℓ j ) e j )) → S (Horn( A, c + ℓ + ( k j − ℓ j + 1) e j ))has a non-trivial kernel. This implies that the space − k j Y k =1 ( h A j , θ i + c j − k ) S (Horn( A, c + ℓ + ( k j − ℓ j ) e j ))is an invariant subspace of S (Horn( A, c + ℓ − ℓ j e j )) . Thus S Horn
A, c + ℓ − P j ∈ J,ℓ j < ℓ j e j !! has an invariant subspace of codimensiongreater than 1. If we consider Y i J,ℓ i < − ℓ i − Y λ i =0 ( h A i , θ i + c i + ℓ i + λ i ) S Horn A, c + ℓ − X j ∈ J,ℓ j < ℓ j e j , it contains a non-trivial monodromy invariant subspace of S Horn A, c + ℓ − X ℓ j < ℓ j e j . Now the proof of the statement is reduced to that for the case ℓ ∈ Z n ≥ . We see that n Y j =1 ℓ j − Y λ j =0 ( h A j , θ i + c j + λ j ) − ( S (Horn( A, c + ℓ )) / Ψ )is an invariant subspace of S (Horn( A, c )) in question. We remark here that none of theoperators h A j , θ i + c j + λ j for j = 1 , . . . , n and λ j = 0 , . . . , ℓ j − P i ( θ ) , Q i ( θ ) , i = 1 , . . . , n of (2.3) for Horn( A, c + ℓ ). (cid:3) AXIMALLY REDUCIBLE MONODROMY OF BIVARIATE HYPERGEOMETRIC SYSTEMS 15
Corollary 3.13.
In the case of two variables, suppose that X A j , A k lin . indep . ν ( A j , A k ) = 0 , where the summation is over all pairs of linearly independent rows of the matrix definingthe Horn system. Then for generic parameter vector c the monodromy representations ofthe Horn systems Horn(
A, c ) and Horn(
A, c − e j ) are equivalent for any j = 1 , . . . , m. Proof.
The condition on the indices of the rows of the defining matrix means precisely (byTheorem 3.7, 3) that there are no persistent polynomial solutions to the Horn system inquestion. Thus for generic parameters all solutions are fully supported (that is, the convexhull of the support of any of the solutions has dimension 2). No such series is annihilatedby a differential operator of the form (3.3) and hence the intertwining operators havetrivial kernels. This means that the monodromy representations are equivalent. (cid:3)
Example 3.14.
The hypergeometric system defined by the matrix(3.4) − − − and the generic parameter vector ( c , c , c ) ∈ C is generated by the differential operators(3.5) (cid:26) x ( θ + 2 θ + c ) + ( θ + θ − c ) ,x ( θ + 2 θ + c )( θ + 2 θ + c + 1) − ( θ + θ − c )( θ − c ) . It is holonomic for any ( c , c , c ) with rank 2. A universal basis in the solution spaceof (3.5), valid for any values of ( c , c , c ) ∈ C , is given by the functions f ( x ; c ) = x c +2 c x − c − c and f ( x ; c ) = x c +2 c ( x − c − c − x c ( x + x + x ) − c − c − c ) / ( c + c + c ) . For c + c + c = 0 , this basis degenerates into the pair of functions x c +2 c x − c − c ,x c +2 c x − c − c log x + x + x x . Observe that the system (3.5) is resonant if and only if c + c + c ∈ Z . The notion of maximal resonance gives nothing new in this example sincethere is only one (up to scaling) linear relation between the rows of the matrix (3.4).Let
Sol ( c ) denote the linear space of local solutions to (3.5) at a nonsingular point. Theintertwining operators for this Horn system are given by I = θ + 2 θ + c − Sol ( c − , c , c ) → Sol ( c ) ,I = − θ − θ + c − Sol ( c , c − , c ) → Sol ( c ) ,I = − θ + c − Sol ( c , c , c − → Sol ( c ) . Observe that I ( f ( x ; c )) = I ( f ( x ; c )) = 0 ,I ( f ( x ; c , c , c − c + c + c − f ( x ; c ) ,I ( f ( x ; c − , c , c )) = I ( f ( x ; c , c − , c )) =( c + c + c ) f ( x ; c ) − f ( x ; c ) ,I ( f ( x ; c , c , c − c + c + c ) f ( x ; c ) . This example shows that the intertwining operators constructed above may have nontrivialkernels despite the fact that the monodromy of (3.5) only depends on the values of c , c , c modulo Z . Explicit monodromy calculation for simplicial and parallelepipedalhypergeometric families
Atomic hypergeometric systems.
In this section, we investigate monodromy rep-resentations of two families of hypergeometric systems. They will generate two classes ofpolygons corresponding to maximally reducible monodromy representations in § Theorem 4.1. (1)
For any × nondegenerate integer matrix M = (cid:18) a b a b (cid:19) and any ˜ c ∈ C the holonomic rank of the associated atomic system is given by rank(Horn( M, ˜ c )) = | det( M ) | + ν ( M ) . Furthermore, there exist | det( M ) | fully supported series solutions of Horn( M, ˜ c ) while the remaining ν ( M ) solutions are persistent Puiseux polynomials. (2) In the case when ν ( M ) > , the initial exponents of the Puiseux polynomial solutionsto Horn( M, ˜ c ) are given by − M − ( R M + ˜ c ) , where R M = ( { ( u, v ) ∈ N : u < | b | , v < | a |} , if | a b | > | b a | , { ( u, v ) ∈ N : u < | a | , v < | b |} , if | a b | < | b a | . Proof. (1) By [18, Proposition 4] the system Horn( M, ˜ c ) admits a solution of the followingform for a suitable cycle C : | det( M ) | (2 πi ) Z C Γ( a s + b s + ˜ c )Γ( a s + b s + ˜ c ) x s x s ds ds = X k ∈ Z ≥ ( − | k | k ! x − M − ( k +˜ c ) = x − M − ˜ c X k ∈ Z ≥ k ! Y j =1 ( − x − M − e j ) k j = x − M − ˜ c exp (cid:18) − X j =1 x − M − e j (cid:19) . (4.1) AXIMALLY REDUCIBLE MONODROMY OF BIVARIATE HYPERGEOMETRIC SYSTEMS 17
The dimension of the linear span of the set of all analytic continuations of (4.1), i.e.the space of the fully supported solutions, equals | det( M ) | sinceG. C. D.(det( M ) , a , b , a , b ) = 1 . By [5, Lemma 6.5] the dimension of the space of persistent Puiseux polynomial solutionsto the system in question is given by ν ( M ). We conclude that rank(Horn( M, ˜ c )) = | det( M ) | + ν ( M ).(2) This statement follows from the construction of persistent Puiseux polynomial so-lutions in [5, Lemma 6.5]. (cid:3) The support of a persistent polynomial solution to a bivariate Horn system can becharacterised as follows. After the above Theorem 4.1, only submatrices A I = ( A i , A j )such that ν ( A i , A j ) > A, ˜ c ) . In makingthe variable change x → x if necessary, we can assume without loss of generality that A i = ( a , b ) ∈ N and A j = ( a , b ) ∈ − N . Furthermore if necessary we change therole of x and x variables to restrict ourselves to the case | a b | > | a b | . In this case R A I = { ( u, v ) ∈ N : u < b , v < | a |} . Corollary 4.2.
Under the above mentioned normalisation setting, we introduce the indexset ˜ R A I = (cid:8) ( u, v ) ∈ N : 0 ≤ u < min( a , b ) , ≤ v < min( | a | , | b | ) (cid:9) , contained in R A I . (1) The support of a persistent monomial solution of the atomic system
Horn( A I , ˜ c I ) is given by α ∈ − A I − ( ˜ R A I + ˜ c I ) . (2) We associate to each α ∈ − A I − (( R A I \ ˜ R A I ) + ˜ c I ) a series of indices S α := S Kk =0 { α k } that will be defined later in the proof.The support of a persistent polynomial solution to Horn( A I , ˜ c I ) is the union of S α andthe supports of persistent monomial solutions.Proof. We first remark that under the above mentioned normalisation, the condition α ∈− A I − ( R A I + ˜ c ) means that P ( α ) = 0 and Q ( α ) = 0. The cardinality of the set of thelattice points satisfying this condition is equal to | a b | .(1) If α ∈ − A I − ( ˜ R A I + ˜ c I ), then α ∈ ker (cid:0) h A i , θ i + ˜ c i + u i (cid:1) ∩ ker (cid:0) h A j , θ i + ˜ c j + v j (cid:1) for ( u i , v j ) ∈ ˜ R A I , and hence the operator h A i , θ i + ˜ c i + u i , u i < min( a , b ) is a factor inboth P ( θ )and P ( θ ). In a similar way h A j , θ i + ˜ c j + v j , v j < min( | a | , | b | ), is a factor inboth Q ( θ ) and Q ( θ ).Let us set i = 1 if ( R A I \ ˜ R A I ) ∩ N × { } 6 = ∅ , and define as usual e = (1 , i = 2 if ( R A I \ ˜ R A I ) ∩ { } × N = ∅ , and e = (0 , | b | < | a | , the case i = 2 arrives. Therefore there exists α such that P ( α ) = Q ( α ) = 0, but Q ( α ) = 0. The following equalities hold: H ( A I , ˜ c I ) x α = ( x P ( θ ) − Q ( θ )) x α = − Q ( α ) x α ,H ( A I , ˜ c I ) x α − e = P ( α − e ) x α − Q ( α − e ) x α − e . Let us now consider the sequence of integer points α , α = α − e , . . . such that α k − α k +1 = − e or e . The points α k lie inside the cone C ( i, j ) := { s : h A j , s i + ˜ c j ≤ } ∩ { s : h A i , s i + ˜ c i ≤ } . This sequence must terminate at a certain step and hencethe union of all points { α k } k ≥ defines a finite subset of C ( i, j ). Thus for a finite setof integer points S α the linear combination of polynomials H ( A I , ˜ c I ) x α k (respectively H ( A I , ˜ c I ) x α k ), k = 1 , . . . , K , is identically equal to zero (see. [5, Lemma 6.5, Fig. 2]depicting the process that is equivalent to the construction of S α ). If | a | ≤ | b | and a ≥ b then ˜ R A I = R A I . Thus all persistent polynomial solutions are actually monomials.If | a | ≤ | b | and a < b , then the case i = 1 arrives. Similarly to the case i = 2,we obtain a polynomial solution supported in the set of integer points S α = S k ≥ { α k } , α = α − e , . . . , such that α k − α k +1 = − e or α k − α k +1 = e . (cid:3) Example 4.3.
The atomic hypergeometric system defined by the matrix M = (cid:18) − − (cid:19) and the zero parameter vector has the form(4.2) x (3 θ + 2 θ )(3 θ + 2 θ + 1)(3 θ + 2 θ + 2) − ( − θ − θ )( − θ − θ + 1)( − θ − θ + 2)( − θ − θ + 3) ,x (3 θ + 2 θ )(3 θ + 2 θ + 1) − ( − θ − θ )( − θ − θ + 1)( − θ − θ + 2) . After Theorem 4.1 (1), the dimension of persistent solutions space is 8.The persistent monomial solutions are given by1 , x − x , x − x , x − x , x − x , x − x . The polynomials x − x − x − x , x − x − x − x + x − x + 12 x − x are the essentially polynomial persistent solutions. We remark here that ( − , ∈− M − (cid:16) R M \ ˜ R M (cid:17) and the solution is binomials in view of | b | − | a | = 1 . Observe that any Puiseux polynomial solution to an atomic system is necessarily per-sistent. This is of course not the case for an arbitrary hypergeometric system.4.2.
Simplicial hypergeometric configurations.
An important special instance of ageneral nonconfluent Horn system is the system defined by a matrix whose rows are thevertices of an n -dimensional integer simplex. More precisely, let M ∈ GL ( n, Z ) be aninteger nondegenerate square matrix and α ∈ C n a parameter vector. Let ˜ α = ( α, α n +1 ) ∈ C n +1 . Denote by M , . . . , M n the rows of the matrix M and let M n +1 = − M − . . . − M n . Let ˜ M be the ( n + 1) × n matrix with the rows M , . . . , M n +1 . The (nonconfluent) Hornsystem Horn( ˜
M , ˜ α ) associated with this data will be called simplicial. Proposition 4.4. (See [17] .) Let us assume that the parameter vector ˜ α is in genericposition. A holonomic simplicial hypergeometric system Horn( ˜
M , ˜ α ) admits the following AXIMALLY REDUCIBLE MONODROMY OF BIVARIATE HYPERGEOMETRIC SYSTEMS 19 solution: (4.3) x − M − α n X j =1 x − M − e j ! −| ˜ α | , where e j = (0 , . . . , , . . . , ( in the j -th position). Any solution to the Horn system Horn( ˜
M , ˜ α ) is either in the linear span of analytic continuations of (4.3) or is a persistentPuiseux polynomial. For −| ˜ α | ∈ N \ { } the monodromy representation of Horn( ˜
M , ˜ α ) ismaximally reducible. Example 4.5.
The Horn system(4.4) (cid:26) x ( θ + θ − θ − θ − − ( − θ + θ )( − θ + θ − ,x ( θ + θ − − θ + θ − − ( θ − θ )( θ − θ − / ( x x ) , x + 2 y + 6 x x + x x + x x ,x − / x − / (5 + 10 x + 30 x x + 20 x x + x x + 5 x x + 10 x x ) ,x − / x − / (5 + 10 x + 30 x x + 20 x x + x x + 5 x x + 10 x x ) . If we consider the Mellin-Barnes integral for the following Ore-Sato coefficient with generic c ∈ R along a proper integration contour C , ϕ ( s ) = Γ( − c + s − s − − s + s − e √− π ( s + s ) Γ( − s − s + 4) , we get a residue that represents a fully supported solution to a Horn system obtained asa perturbation of (4.4) i.e. the result of replacement of θ − θ by θ − θ − c : f c = x − c − x − c − (cid:16) x / x / + x / x / + 1 (cid:17) − c . Observe that for c = 0 we get the Puiseux polynomial solution, f = (cid:16) x / x / + x / x / + 1 (cid:17) x x . The reason for this phenomenon lies in the fact that in ϕ ( s ) the poles of the numeratorΓ( − c + s − s −
1) are not cancelled by those of the denominator Γ( − s − s + 4) forgeneric c. For c = 0 the half-space cancellation of poles (see Definition 6.2) happens andthe poles are located in the strip { s : − ≤ s + s ≤ } .Linear combinations of several analytic continuations of f produce three Puiseux poly-nomial solutions to (4.4) except the first one. The only persistent solution in this exampleis the Laurent monomial 1 / ( x x ) ∈ ker ( θ − θ − ∩ ker ( − θ + θ − with respect tothe monodromy action. ✻ ✲ st ❅❅❅❅❅❅❅❅❅❅❅❅❅❅✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟ ❢ rr rrr rt ttt tt t❞❞ ❞ ❞❞ ❞❞ a): The supports of solutions to (4.4) ✻ ✲✁✁✁✁✟✟✟✟❅❅ rr rr b): The polygon of the Ore-Satocoefficient defining (4.4) Figure 1.
Example 4.6.
Let us consider the bivariate ( n = 2) simplicial hypergeometric systemgenerated by the matrix M = (cid:18) − − (cid:19) and the vector of parameters ˜ α = (0 , , c ) in the sense of the definition in the beginningof this subsection. This choice of the parameters does not affect the generality of thepresent example since changing the first two coordinates of ˜ α only results in a shift of theexponent space. This system is generated by the differential operators(4.5) (cid:26) x (2 θ + 2 θ + c )(2 θ + 2 θ + c + 1) − θ (2 θ − ,x (2 θ + 2 θ + c )(2 θ + 2 θ + c + 1) − θ (2 θ − . By Theorem 3.8 the holonomic rank of (4.5) equals 4. By Proposition 4.4 the generatingsolution to (4.5) is given by (1+ √ x + √ x ) − c . It follows from Theorem 3.7 that (4.5) doesnot admit any persistent Puiseux polynomial solutions and therefore for generic c ∈ C abasis in the space of analytic solutions to (4.5) is given by(4.6) f ( c ) = (1 + √ x + √ x ) − c ,f ( c ) = (1 + √ x − √ x ) − c ,f ( c ) = (1 − √ x + √ x ) − c ,f ( c ) = (1 − √ x − √ x ) − c . However, this basis degenerates for two special values of c, namely for c = 0 (when all thebasis elements (4.6) are identically equal to 1) and for c = − f ( − − f ( − − f ( −
1) + f ( − ≡ c. If c = − , the corresponding resonant basis is given by f ( − , f ( − , f ( −
1) and thefunction ˜˜ f = ( f log f − f log f − f log f + f log f ) (cid:12)(cid:12) c = − . AXIMALLY REDUCIBLE MONODROMY OF BIVARIATE HYPERGEOMETRIC SYSTEMS 21
For c = 0 , a basis in the solution space of (4.5) is given by f (0) and the three additionalresonant solutions ˜ f = log(1 + √ x + √ x ) − log(1 + √ x − √ x ) , ˜ f = log(1 + √ x + √ x ) − log(1 − √ x + √ x ) , ˜ f = log(1 + √ x + √ x ) − log(1 − √ x − √ x ) . However, it turns out to be possible to construct a single universal basis in the space ofanalytic solutions to (4.5) whose elements remain linearly independent after passing tothe limit as c → c → − . This basis has the following form:(4.7)ˆ f ( c ) = (cid:0) √ x + √ x (cid:1) − c , ˆ f ( c ) = (cid:0) (1 + √ x + √ x ) − c − (1 + √ x − √ x ) − c (cid:1) /c, ˆ f ( c ) = (cid:0) (1 + √ x + √ x ) − c − (1 − √ x + √ x ) − c (cid:1) /c, ˆ f ( c ) = (cid:0) (1 + √ x + √ x ) − c − (1 + √ x − √ x ) − c − (1 − √ x + √ x ) − c + (1 − √ x − √ x ) − c (cid:1) / ( c + c ) . It is easy to check that the functions ˆ f ( c ) , . . . , ˆ f ( c ) are linearly independent for any c ∈ C . Given the basis (4.7), it is straightforward to find the monodromy representation of thefundamental group of the complement to the singularities of the solutions to (4.5). It isgenerated by three matrices corresponding to the loops around the coordinate axes { x =0 } , { x = 0 } and the essential singularity {S ( x ) := 1 − x + x − x − x x + x = 0 } . These matrices are given by M x = − c
00 1 0 − − c − − , M x = − c − − − c − ,M S = diag( e − π √− c ) . Parallelepipedal hypergeometric configurations.
Let M ∈ GL ( n, Z ) be aninteger nondegenerate square matrix and let α, β ∈ C n be two parameter vectors. Denoteby ˜ M the 2 n × n matrix obtained by joining together the rows of the matrices M and − M. The rows of such a matrix define the vertices of a parallelepiped of nonzero n -dimensionalvolume. Let ˜ α be the vector with the components ( α , . . . , α n , β , . . . , β n ) . It turns outthat the corresponding Horn system Horn( ˜
M , ˜ α ) admits a simple basis of solutions. Proposition 4.7. (See [18] .) Let us assume that the parameter vector ˜ α is in genericposition. The holonomic hypergeometric system Horn( ˜
M , ˜ α ) admits the following solution: (4.8) x − M − α n Y j =1 (cid:16) x − M − e j (cid:17) − α j − β j , where e j = (0 , . . . , , . . . , ( in the j -th position). Any solution to the hypergeometricsystem Horn( ˜
M , ˜ α ) is either in the linear span of analytic continuations of (4.8) or is a persistent Puiseux polynomial. If − α j − β j ∈ N \ { } for any j = 1 , . . . , n then themonodromy representation of Horn( ˜
M , ˜ α ) is maximally reducible. Bases in the solution space of the Horn system
Let us denote by q the number of vertices of the Newton polytope of the polynomialwhich defines the singular hypersurface of the hypergeometric system under study. In thissection we construct a family of q bases in the space of fully supported solutions to thathypergeometric system. This result will be used in Section 6 to deduce the main result ofthe paper. Definition 5.1.
The amoeba A f of a Laurent polynomial f ( x ) (or of the algebraic hy-persurface f ( x ) = 0) is defined to be the image of the hypersurface f − (0) under the mapLog : ( x , . . . , x n ) (log | x | , . . . , log | x n | ) . Let A ( ϕ ) denote the amoeba of the singularity of the hypergeometric system Horn( ϕ ) . Definition 5.2.
For a convex set B ⊂ R n its recession cone C B is defined to be C B = { s ∈ R n : u + λs ∈ B, ∀ u ∈ B, λ ≥ } . That is, the recession cone of a convex set is themaximal element (with respect to inclusion) in the family of those cones whose shifts arecontained in this set.The following theorem (cf. the results in [9] for the Gelfand-Kapranov-Zelevinsky sys-tem) shows that for any vertex of the Newton polygon of the singularity of a bivariatehypergeometric function there exists a basis in the solution space of the correspondingHorn system. This basis consists of hypergeometric series which converge on the preimageof the amoeba complement which corresponds to that vertex. Theorem 5.3. (1)
For any bivariate nonconfluent Ore-Sato coefficient ϕ with generic pa-rameters and any connected component M of c A ( ϕ ) there exists a pure Puiseux series basis f M,i , i = 1 , . . . , rank(Horn( ϕ )) in the solution space of Horn( ϕ ) such that the recessioncone of the support of f M,i is contained in − C ∨ M . (2) The domain of convergence of the series f M,i contains
Log − ( M ) for any i =1 , . . . , rank(Horn( ϕ )) . Proof.
Let the Ore-Sato coefficient defining the Horn system be of the form ϕ ( s ) = m Y i =1 Γ( a i s + b i s + c i ) , where ( a i , b i ) ∈ Z , P mi =1 ( a i , b i ) = (0 ,
0) and c = ( c , . . . , c m ) ∈ C m is a generic parametervector. By Theorem 2 in [16] the vectors { ( a i , b i ) } mi =1 are the normals to all sides of thepolygon P ( ϕ ) of the Ore-Sato coefficient ϕ (observe that some of them may coincide).This theorem also implies that the number of different vectors in this set equals q. Tosimplify the notation, we denote the different elements in this set of outer normals to P ( ϕ )by ( a , b ) , . . . , ( a q , b q ) . We may without loss of generality assume that these normals areordered counterclockwise from ( a , b ) to ( a m , b m ) . Let v i denote the vertex of P ( ϕ ) thatjoins the sides with the normals ( a i , b i ) and ( a i +1 , b i +1 ) ( v m being the vertex that joinsthe first and the last sides of the polygon). By Theorem 7 in [12] there is a one-to- AXIMALLY REDUCIBLE MONODROMY OF BIVARIATE HYPERGEOMETRIC SYSTEMS 23 M M M M M M M Figure 2.
The amoeba of the singularity of a Horn system one correspondence between the vertices v , . . . , v q and the connected components of thecomplement of A ( ϕ ) . Let M , . . . , M q be the connected components of the complementof A ( ϕ ) . In Figure 2 we depict the special case of the amoeba of the singularity of the Hornsystem defined by the Ore-Sato coefficient Γ( s + 2 s )Γ( s − s )Γ( − s + 3 s )Γ( − s − s )Γ( s )Γ( − s − s )Γ( s ) . In this case q = 7 . The continuous curve that bounds theamoeba and goes inside is its contour (see [13]). The shape of the amoeba was foundby means of the Horn-Kapranov parametrisation ([20]) using computer algebra systemMathematica 9.0. Figure 2 also shows the recession cones of the convex hulls of theconnected components of the amoeba complement that are strongly convex and contain M . The duals of these cones support hypergeometric series whose domains of convergencecontain Log − M . To prove the theorem, we need to show that the number of such seriesis independent of the connected component of the amoeba complement.Let us prove that for any i = 1 , . . . , q the number of fully supported Puiseux seriessolutions to Horn( ϕ ) which converge on Log − ( M i ) is the same. To prove this, we willshow that the number of such series whose domain of convergence is Log − ( M ) coincideswith the number of Puiseux series solutions that converge on Log − ( M ) . Repeating thisargument, one can prove that for any two adjacent components in the complement of A ( ϕ )the number of Puiseux series solutions that converge on preimages of these componentsunder the map Log is the same. This will prove that any such connected componentcarries the same number of fully supported Puiseux series solutions.Let us define the single-valued branch arg of the argument function Arg by settingarg( − a − b √−
1) = 0 , and lim ε → − arg e √− ε ( − a − b √−
1) = 2 π. We introduce the partialorder ≺ on Z by saying that ( a, b ) ≺ ( c, d ) if arg( a + b √− < arg( c + d √− . We willsay that ( a, b ) ( c, d ) if arg( a + b √− ≤ arg( c + d √− . By Lemma 11 in [12] and Theorem 4.1 the number of fully supported Puiseux seriessolutions to the hypergeometric system Horn( ϕ ) that converge in the domain Log − ( M i )equals S i = X j : − (¯ a i +1 , ¯ b i +1 ) ≺ (¯ a j , ¯ b j ) (¯ a i , ¯ b i ) ,ℓ : (¯ a i +1 , ¯ b i +1 ) (¯ a ℓ , ¯ b ℓ ) ≺− (¯ a j , ¯ b j ) k j k ℓ (cid:12)(cid:12)(cid:12)(cid:12) ¯ a ℓ ¯ b ℓ ¯ a j ¯ b j (cid:12)(cid:12)(cid:12)(cid:12) , where k j is the number of elements in the set of vectors { ( a , b ) , . . . , ( a m , b m ) } ,that co-incide with (¯ a j , ¯ b j ). Observe that by our choice of the indices of summation all of theinvolved determinants are positive. To prove that S = S we make use of the fact thatthese two sums have many common terms. Indeed, the sum of terms in S that are notpresent in S is given by(5.1) X j : − (¯ a , ¯ b ) ≺ (¯ a j , ¯ b j ) (¯ a , ¯ b ) k k j (cid:12)(cid:12)(cid:12)(cid:12) ¯ a ¯ b ¯ a j ¯ b j (cid:12)(cid:12)(cid:12)(cid:12) = det (cid:18) k (¯ a , ¯ b ) , X j : − (¯ a , ¯ b ) ≺ (¯ a j , ¯ b j ) (¯ a , ¯ b ) k j (¯ a j , ¯ b j ) (cid:19) . Similarly, the sum of terms in S that are not present in S is given by(5.2) X ℓ : (¯ a , ¯ b ) (¯ a ℓ , ¯ b ℓ ) ≺− (¯ a , ¯ b ) k k ℓ (cid:12)(cid:12)(cid:12)(cid:12) ¯ a ℓ ¯ b ℓ ¯ a ¯ b (cid:12)(cid:12)(cid:12)(cid:12) = det (cid:18) X ℓ : (¯ a , ¯ b ) (¯ a ℓ , ¯ b ℓ ) ≺− (¯ a , ¯ b ) k ℓ (¯ a ℓ , ¯ b ℓ ) , k (¯ a , ¯ b ) (cid:19) . The nonconfluency condition P qi =1 k i (¯ a i , ¯ b i ) = P mj =1 ( a j , b j ) = (0 ,
0) implies that the deter-minant in the right-hand side of (5.1) equals the determinant in right-hand side of (5.2).This proves that any connected component of the amoeba complement carries equallymany fully supported solutions to the Horn system.It remains to observe that any solution of a hypergeometric system with generic pa-rameters can be expanded into a Puiseux series with the center at the origin. (This seriesmay turn out to be a Puiseux polynomial.) Since a Puiseux polynomial solution to aHorn system is defined everywhere except (possibly) the coordinate hyperplanes, it worksfor any connected component in the complement of the amoeba of the singularity. Thusfor any such component M there exists a Puiseux series basis in the solution space of theHorn system all of whose elements converge (at least) in the domain Log − ( M ) . Now we see that we can take pure Puiseux series as a basis. For this purpose we showthat suitable linear combinations of the analytic continuation of a solution P ( x ) = µ X k =1 x v kN x v kN p k ( x , x )where p k ( x ) , k = 1 , . . . , µ are power series that converge in Log − ( M i ) for a fixed i,N , N ∈ N , v k , v k ∈ Z . It is worthy noticing that µ ≤ N · N . The result of an analyticcontinuation along the loop turning around ℓ times around x = 0 and ℓ times around x = 0 will be( M ℓ x =0 M ℓ x =0 ) ∗ P ( x ) = µ X k =1 e (cid:16) ℓ v kN + ℓ v kN (cid:17) π √− x v kN x v kN p k ( x , x ) . AXIMALLY REDUCIBLE MONODROMY OF BIVARIATE HYPERGEOMETRIC SYSTEMS 25
To obtain x v kN x v kN p k ( x , x ) as a linear combination of ( M ℓ x =0 M ℓ x =0 ) ∗ P ( x ) , ≤ ℓ ≤ N − , ≤ ℓ ≤ N − µ. This completes the proof of the theorem. (cid:3) Maximally reducible monodromy
In this section we restrict our attention to bivariate Horn systems. Let A be an integer m × A the convex polygon P withinteger vertices such that the outer normals to the sides of P are the rows of A. We alsorequire that the relative length of a side of P in the integer lattice equals the number ofoccurrences of the corresponding (normal) row in the matrix A. (Observe that the normalsto a polygon whose lengths are adjusted in this way sum up to zero.) According to theMinkowski theorem the polygon P satisfying these conditions is uniquely determined (upto a translation by an integer vector) by the matrix A. Conversely, any plane convexinteger polygon P defines the matrix A ( P ) whose rows are the outer normals to its sides(with some of them possibly repeated). The order of the rows of this matrix is unimportantsince they all lead to the same hypergeometric system of equations. Thus, together withthe vector of parameters c , such a polygon defines a nonconfluent hypergeometric systemof equations which we denote by Horn( A ( P ) , c ) . This has been illustrated by Example 4.5.The results of Section 4 yield that any Horn system defined by a matrix whose rowsare the vertices of a simplex or a parallelepiped admits a basis of Puiseux polynomials forsuitable values of its parameters. In particular, the monodromy representation of such aHorn system (with this very particular choice of parameters) is maximally reducible.In the paper [4] the authors have posed the problem of describing the Gelfand-Kapranov-Zelevinsky hypergeometric systems (see [9]), whose solution space contains a one-dimen-sional subspace with the trivial action of monodromy on it. (This corresponds to theexistence of a rational solution.) In the present section, we will resolve the closely relatedproblem of describing the class of Horn hypergeometric systems with maximally reduciblemonodromy representations. Apart from systems with rational bases of solutions, suchsystems have the simplest possible monodromy representation since the correspondingmonodromy groups are generated by diagonal matrices.Recall that a zonotope is the Minkowski sum of segments. The main result in thissection is the following theorem.
Theorem 6.1.
The monodromy representation of a bivariate nonconfluent hypergeometricsystem
Horn( A ( P ) , c ) is maximally reducible for some c ∈ C n if and only if the polygon P is either 1) a zonotope; or 2) the Minkowski sum of a triangle △ and an arbitrary numberof segments that are parallel to the sides of △ . For instance, the zonotope in Figure 6 corresponds to the matrix (6.9) whose rows arethe outer normals to its sides.
Theorem 6.1 implies that any triangle defines a hypergeometric system with a maxi-mally reducible monodromy (for a suitable choice of the vector of parameters). A quadri-lateral defines a system with a maximally reducible monodromy if and only if it is atrapezoid.We divide the proof of Theorem 6.1 into three steps.We first give a detailed description of a key technical notion named ”half-space cancel-lation of poles” (Definition 6.2, Lemma 6.3). Then we prove that each of the conditions1),2) is necessary and sufficient for the conclusion of the theorem to hold (Propositions6.5, 6.6). Finally we establish the fact that the maximal reducibility of the monodromy isequivalent to the existence of a Puiseux polynomial basis for a proper choice of parameters(Corollary 6.7) with the aid of Proposition 6.6.To prove the necessity and sufficiency of the condition in Theorem 6.1 we will need thefollowing auxiliary technical notion.
Definition 6.2.
We will say that the Ore-Sato coefficient ϕ ( s ) = Q aj =1 Γ( α j ) Q bi =1 Γ( β i ) admits a half-space cancellation of poles, if the poles of ϕ ( s ) lie in the set { s : α j ( s ) = σ, σ ∈ Z ≤ , γ j ≤ σ ≤ } for some γ j < , j ∈ [1 , a ] . Lemma 6.3.
The half-space cancellation of poles in the Ore-Sato coefficient ϕ ( s ) = Q aj =1 Γ( α j ) Q bi =1 Γ( β i ) is a necessary condition for the Mellin-Barnes integral MB( ϕ, C ) to presenta set of Puiseux polynomial solutions for every contour C , satisfying the conditions inTheorem 3.3. Example 6.4.
Consider the function ϕ ( s ) = Γ( s + s − − s )Γ( s + 1)Γ( s + 2)Γ( − s + 2) . Its poles are located on the lines { s : − s = σ, σ = − , , s = − , − , . . . } . In thiscase MB( ϕ, C ) = const · ( x + 1) (2 x − x + 2) , where the contour C is located aroundthe integer lattice points inside of { s : s + s ≤ , ≤ s , ≤ s } . We now make use of Definition 6.2 and Lemma 6.3 to prove the sufficiency of either orthe conditions 1), 2).
Proposition 6.5.
For a polygon P of type 1) or 2), Horn( A ( P ) , c ) admits a Puiseuxpolynomial basis for some parameter c ∈ C n and hence admits a maximally reduciblemonodromy representation.Proof. Let A be a m × c ∈ C m such that the space of holomorphic solutions to the hypergeometric systemHorn( A, c ) at a generic point has a basis that consists of functions of the form x α p ( x ) , where α ∈ C n , and p ( x ) is a (Taylor) polynomial. Since the analytic continuation of sucha function along any path is proportional to itself, this will prove that the monodromyrepresentation of Horn( A, c ) is maximally reducible.Since the matrix A defines a zonotope, we may without loss of generality assume(possibly after interchanging some of its rows) that it consists of blocks of the form AXIMALLY REDUCIBLE MONODROMY OF BIVARIATE HYPERGEOMETRIC SYSTEMS 27 B i = (cid:18) a i b i − a i − b i (cid:19) . Let us denote by k i the number of occurrences of the block B i in thematrix A and let l denote the number of different blocks. By Theorem 3.8 the holonomicrank of the system Horn( A, c ) equals r ( A ) = l X i =1 k i | a i | ! l X j =1 k j | b j | ! − l X i =1 k i | a i b i | = l X i, j = 1 i = j k i k j | a i b j | . We will use induction with respect to l to show that the hypergeometric system Horn( A, c )admits a Puiseux polynomial basis in the linear space of its analytic solutions. For l = 2we have a parallelogram which by Proposition 4.7 (for − α j − β j ∈ N \ { } in (4.8) )defines a system with a Puiseux polynomial basis in its solution space.Let the matrix be defined through B l +1 = (cid:18) a l +1 b l +1 − a l +1 − b l +1 (cid:19) .Denote by A ′ the matrix that is obtained by appending k l +1 copies of the block B l +1 to the matrix A and let r ( A ′ ) denote the holonomic rank of the associated Horn system.Similarly to the above, we may without loss of generality assume that a l +1 = 0, b l +1 = 0.We may also assume that the vector ( a l +1 , b l +1 ) is not proportional to ( a i , b i ) for any i = 1 , . . . , l . For if these two vectors were proportional, adding the block B l +1 would beequivalent to increasing the number k i of occurrences of the block B i in the matrix A .Observe that appending the block B l +1 to the matrix A corresponds to adding thesegment ( − b l +1 , a l +1 ) by Minkowski to the polygon that is defined by the matrix A. Inthis case, the amoeba of the singularity of the corresponding hypergeometric systemssprouts two new tentacles in opposite directions. This can be seen from [12], Lemma 11(two-sided Abel’s lemma). By Theorem 5.3 the number of Puiseux series solutions is thesame for every connected component of its complement. We will show that for a suitable(and, of course, a very specific) choice of the parameters of the system these series actuallyturn out to be polynomials.Under the above assumptions the holonomic rank r ( A ′ ) of the hypergeometric systemdefined by the matrix A ′ and a generic vector of parameters is given by r ( A ′ ) = l +1 X i, j = 1 i = j k i k j | a i b j | = r ( A ) + l X i =1 k i k l +1 | a i b l +1 | + l X j =1 k l +1 k j | a l +1 b j | = r ( A ) + l X i =1 (cid:0) ( k i | a i | + k l +1 | a l +1 | )( k i | b i | + k l +1 | b l +1 | ) − k i | a i b i | − k l +1 | a l +1 b l +1 | (cid:1) = r ( A ) + l X i =1 r ( k i B i , k l +1 B l +1 ) , where r ( k i B i , k l +1 B l +1 ) stands for the holonomic rank of the parallelepipedal hypergeo-metric system defined by the matrix obtained by joining together k i copies of the block B i and k l +1 copies of the block B l +1 . Using Proposition 4.7 and Theorem 5.3 we conclude that adding (by Minkowski) a segment to a plane zonotope preserves the property of thecorresponding hypergeometric system to have a Puiseux polynomial basis in its space ofholomorphic solutions (for a suitable choice of the vector of parameters).We first observe that for some positive integer m l +1 the poles of the meromorphicfunction Γ( a l +1 s + b l +1 s + c l +1 )Γ( a l +1 s + b l +1 s + c l +1 + m l +1 + 1)are located on the lines m l +1 S h =0 { s : a l +1 s + b l +1 s + c l +1 + h = 0 } . The poles of the function l Y i =1 k i Y j =1 Γ( a i s + b i s + c i,j )Γ( a i s + b i s + c i,j + m i,j + 1)are also located in the finite family of lines S li =1 S k i j =1 S m i,j h =0 { s : a i s + b i s + c i,j + h = 0 } .We conclude that for a suitable choice of the vector of parameters c the number of doublepoles of the meromorphic function l +1 Y i =1 k i Y j =1 Γ( a i s + b i s + c i,j )Γ( a i s + b i s + c i,j + m i,j + 1)is finite. To prove this fact it suffices to choose the vector of parameters c so that theparallelogramΠ( i, j ; k, ℓ ) = [ t =0 1 [ u =0 { s : a i s + b i s + c i,j + tm i,j = 0 , a k s + b k s + c k,ℓ + um k,ℓ = 0 } does not intersect any similar parallelogram Π( i ′ , j ′ ; k ′ , ℓ ′ ), as long as | i − i ′ | + | j − j ′ | + | k − k ′ | + | ℓ − ℓ ′ | 6 = 0. Remark that all double poles of the meromorphic functionΓ( a i s + b i s + c i,j )Γ( a k s + b k s + c k,ℓ )Γ( a i s + b i s + c i,j + m i,j + 1)Γ( a k s + b k s + c k,ℓ + m k,ℓ + 1) , that contribute to the solutions of Horn( A, c ), are contained in Π( i, j ; k, ℓ ) the parallel-ogram defined above thanks to the cancellation of poles (cf. Definition 6.2) of the twofactors Γ( a i s + b i s + c i,j ) and Γ( a k s + b k s + c k,ℓ ). Since a parallelogram is the im-age of the square { ( t, u ) : 0 ≤ t ≤ , ≤ u ≤ } under a linear map, it is possible tochoose values of the parameters c i,j , c k,ℓ , c i ′ ,j ′ , c k ′ ,ℓ ′ so that Π( i, j ; k, ℓ ) does not inter-sect Π( i ′ , j ′ ; k ′ , ℓ ′ ) for ( i, j ; k, ℓ ) = ( i ′ , j ′ ; k ′ , ℓ ′ ). The set of such pairs is finite and thereforethe desired choice of parameters can always be made.The inductive step described above is illustrated by Figure 3 under the assumption that a i , b i > i = 1 , ,
3. The shaded regions contain the supports of Puiseux polynomialsolutions to the Horn system obtained by adding the block B = (cid:18) a b − a − b (cid:19) to thehypergeometric system defined by the matrix composed of the blocks B and B . Theabove rank computation shows that the Puiseux polynomial solutions emerging at theintersections of the new (the third) pair of divisors with the initial divisors is exactlysufficient to compensate the rank growth. In fact, by Theorem 3.8 the rank of the system
AXIMALLY REDUCIBLE MONODROMY OF BIVARIATE HYPERGEOMETRIC SYSTEMS 29 defined by all three pairs of divisors equals ( a + a + a )( b + b + b ) − a b − a b − a b . This is exactly how many Puiseux polynomials are supported by the three parallelogramsdepicted in Figure 3. ❅❅❅❅❅❅❅❅❅❅❅❅❅❅ ❅❅❅❅❅❅❅❅❅❅❅❅❅❅ (cid:0)✒ ( a , b ) (cid:0)✠ ( − a , − b ) (cid:0)✠ a b + a b solutions supported here ❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈ ❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈ ✘✘✾ ( − a , − b ) ✘✘✿ ( a , b ) ✛ a b + a b solutions supported here PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP✂✂✍ ( a , b ) ✂✂✌ ( − a , − b ) (cid:0)(cid:0)(cid:0)✠ a b + a b solutions supported here r r r r . . . .. . . .. . .. . .. .. .. .... .... .. .. . .. . .. . . r r r r . . . . .. . . .. . . .. . .. . .. . .. .. ... . .. .. . .. . . . rr r r . . . .. . . .. . .. . .. . .. .. .. .. . Figure 3.
Adding a segment to a zonotope that defines a Horn system
Similar arguments show that the second class of polygons in Theorem 6.1 (the sums oftriangles in the sense of Minkowski and multiples of their sides) also define hypergeometricsystems with Puiseux polynomial bases.Since any pure Puiseux polynomial spans a one-dimensional invariant subspace, it fol-lows that the monodromy representation of a hypergeometric system satisfying the con-ditions of Theorem 6.1 is maximally reducible. (cid:3)
Now we prove the necessity of the conditions 1), 2) of Theorem 6.1.
Proposition 6.6.
If a hypergeometric system Horn(
A, c ) has a maximally reducible mon-odromy representation then its Ore-Sato polygon must be either 1) a zonotope or 2) theMinkowski sum of a triangle and segments parallel to the sides of it.Proof.
To simplify the exposition we treat the case where the matrix A has the followingform:(6.1) A ′ = a b . . . . . .a r b r , where 1 + P rj =1 a j = 1 + P rj =1 b j = 0, m = r + 2. The proof for the general form of A can be achieved in a completely parallel way.As a triangle Ore-Sato polygon means condition 2) case, in the cases that interest usfurther the number r shall be greater than 2 so that m ≥ . Further we shall use thenotation α j ( s ) = a j s + b j s . We consider two groups of linear functions α j ( s ) that areindexed by I + , I − in such a way that j + ∈ I + (resp. k − ∈ I − ) if and only if a j + > a k − < α j + ( s )+ γ j + ), α j + ( s ) = − m − γ j + , m ∈ Z ≥ (resp. Γ( α k − ( s ) + γ k − ), α k − ( s ) = − m − γ k − , m ∈ Z ≥ ) restricted to the complex plane { s ∈ C : s + δ + n = 0 , n ∈ Z ≥ } behave like s → −∞ (resp. s → + ∞ ).For the function(6.2) ϕ ,j + ,k − ( s ) = Γ( s + δ )Γ( α j + ( s ) + γ j + )Γ( α k − ( s ) + γ k − )Γ(1 − s − δ ) Q rℓ = j + ,k − Γ(1 − α ℓ ( s ) − γ ℓ )we examine the solution subspace of S (Horn( A ′ , c ′ )) spanned by u ,j + ( x ) = 1(2 π √− Z C ,j + ϕ ,j + ,k − ( s ) x s ds, and its analytic continuations. Here c ′ = ( δ , δ , γ , . . . , γ r ) and C ,j + = { s ∈ C : | s + δ + n | = | α j + ( s ) + γ j + + m | = ε, ( n, m ) ∈ Z ≥ } . The circle radius ε is chosen to be small enough so that each disk inside the circle containsone isolated double pole of ϕ ,j + ,k − ( s ).We remark that the space of solutions to a resonant system Horn( A ′ , c ′ ) (see Defini-tion 2.13) has a non-diagonalisable monodromy representation except in the trivial caseof a system of holonomic rank 1. That is, for such a system at least one of the mon-odromy representation matrices would have a non-trivial Jordan cell of size at least 2.Thus already it is not maximally reducible. Therefore we may assume that Horn( A ′ , c ′ )is non-resonant. This means that the solution u ,j + ( x ) can be expanded into the Puiseuxseries(6.3) X ( n,m ) ∈ Z ≥ c n,m x bj + aj + x n + δ x − m − γj + aj + , in the neighbourhood of ( x , x ) = (0 , . Repeated application of the monodromy action x → e π √− x to the above series representation of u ,j + ( x ) produces a j + -dimensionalsubspace S ,j + ⊂ S (Horn( A ′ , c ′ )) due to the non-degeneracy of a Vandermonde matrix.Now we consider the analytic continuation of the Puiseux series solution u ,j + ( x ) (6.3)to(6.4) u ,k − ( x ) = 1(2 π √− Z C ,k − ϕ ,j + ,k − ( s ) x s ds, by means of the Mellin-Barnes contour throw (See Fig. 4).The above integral is calculated as the residue along the contours C ,k − = { s ∈ C : | s + δ + n | = | α k − ( s ) + γ k − + m | = ε, n, m ∈ Z ≥ } , that encircle poles on the complex plane { s ∈ C : s + δ + n = 0 , n ∈ Z ≥ } such that s → + ∞ . The Puiseux expansion of u ,k − ( x ) in the neighbourhood of (cid:16) x , x (cid:17) = (0 , AXIMALLY REDUCIBLE MONODROMY OF BIVARIATE HYPERGEOMETRIC SYSTEMS 31 ✲ ☛ ✟q ☛ ✟q ☛ ✟q q q q❛ ❛ ❛ ❛ ❛☛ ✟☛ ✟☛ ✟ ✛ q q q q q q❛ ❛ ❛ ❛ ❛✡ ✠✡ ✠✡ ✠✡ ✠✡ ✠✡ ✠ ✛✲ ❛ : α j + ( s ) = 0 , − , − , . . . q : α k − ( s ) = 0 , − , − , . . . ❈❈❈❈❈❈❈❈❲ Mellin-Barnes contour throw
Figure 4.
Mellin-Barnes contour throwhas the following form: X ( n,m ) ∈ Z ≥ d n,m x bk − ak − x n + δ x − m − γk − ak − , with a k − < . Repeated application of the monodromy action x → e π √− x to the aboveseries presentation of u ,k − ( x ) produces | a k − | -dimensional subspace S ,k − of the solutionspace S (Horn( A ′ , c ′ )) due to non-degeneracy of a Vandermonde matrix.Now we analyse the following analytic continuation steps:a) The analytic continuation of u ,j + to S ,k − by Mellin-Barnes contour throw.b) Monodromy action on S ,k − induced by the map x e πh √− x , i.e. ϕ ,j + ,k − ( s ) x s ϕ ,j + ,k − ( s ) e πhs √− x s , h ∈ Z . c) Inverse analytic continuation of S ,k − to S ,j + . Under the condition of the maximal reducibility of monodromy, if the above proceduresa), b), c) give rise to a well-defined non-trivial monodromy around x = ∞ , the imageof S ,j + under this monodromy action has dimension | a k − | and hence | a j + | = | a k − | . Thismeans that for every j + ∈ I + , there exists k − ∈ I − such that a j + + a k − = 0 . We can apply the same argument in changing the role of s and s , i.e. x and x in (6.3),(6.4) to conclude that for every b p + > b q − < b p + + b q − = 0 . Now we show a stronger assertion than the one that has been shown: for every j + ∈ I + , there exists k − ∈ I − such that(6.5) a j + + a k − = 0 , b j + + b k − = 0 . To prove the existence of such an index, we study the convergence domain of every possibleseries defined as a residue of ϕ i,j + ,k − ( s ) x s .Let us denote by D j + ,k − the convergence domain of the series u j + ,k − ( x ) = X n,m ≥ Res α j + ( s ) + γ j + = − n,α k − ( s ) + γ k − = − m ϕ i,j + ,k − ( s ) x s , for i = 1 , j + ∈ I + , k − ∈ I − . Here we used the notation ϕ ,j + ,k − ( s ) = Γ( s + δ )Γ( α j + ( s ) + γ j + )Γ( α k − ( s ) + γ k − )Γ(1 − s − δ ) r Q ℓ = j + ,k − Γ(1 − α ℓ ( s ) − γ ℓ ) . In a similar way, we look at the convergence domains D i,j + of the series u i,j + ( x ) = X n,m ≥ Res α j + ( s ) + γ j + = − m,s i + δ i = − n ϕ i,j + ,k − ( s ) x s , and D i,k − of the series u i,k − ( x ) = X n,m ≥ Res α k − ( s ) + γ k − = − m,s i + δ i = − n ϕ i,j + ,k − ( s ) x s , for i = 1 , . Now we will establish the following statement: D j + ,k − has a nonempty intersection withat least one of the four domains D ,j + , D ,j + , D ,k − , D ,k − . To prove this claim we consider the supporting cones C j + ,k − , C i,j + , and C i,k − of thesolutions u j + ,k − ( x ) , u i,j + ( x ) , and u i,k − ( x ) respectively. The Abel lemma ([9] Proposition2, [12] Lemma 1) implies the inclusionLog x ( a,b ) − C ∨ a,b ⊂ Log ( D a,b )for some x ( a,b ) ∈ D a,b and ( a, b ) = ( j + , k − ) or ( i, j +) or ( i, k − ) . After an easy case bycase study we see that C ∨ j + ,k − has nonempty two dimensional intersection with one of fourdual cones C ∨ ,j + , C ∨ ,j + , C ∨ ,k − , C ∨ ,k − . This proves the claim ( See Figure 5).Let us assume, for example, D j + ,k − ∩ D ,j + = ∅ . The analytic continuation of S ,j + induced by a Mellin-Barnes throw C ,j + → C j + ,k − on the complex planes { s ∈ C : α j + ( s ) + γ j + ∈ Z ≤ } produces a | a j + ( b j + + b k − ) | -dimensional Puiseux series solutionsubspace of S (Horn( A ′ , c ′ )) convergent on D j + ,k − by virtue of Theorem 3.7 (2). Thisdimension is calculated by the following equalities,(6.6) (cid:12)(cid:12)(cid:12)(cid:12) det (cid:18) a j + b j + a k − b k − (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) = | a j + ( b j + + b k − ) | , where a j + = − a k − . On the other hand, we had already noticed that the analytic con-tinuation S ,k − of S ,j + induced by the Mellin-Barnes contour throw C ,j + → C ,k − onthe complex planes { s ∈ C : s + δ ∈ Z ≤ } has dimension | a k − | = a j + . Thus we ob-tained an analytic continuation of S ,j + convergent on D j + ,k − ∩ D ,j + = ∅ with dimension a j + + | a j + ( b j + + b k − ) | by Theorem 3.7 (2). If the monodromy is maximally reducible, thenevery analytic continuation of S ,j + , including the results of monodromy actions, musthave dimension a j + . This means that b j + + b k − = 0 and hence (6.5) follows.If D j + ,k − ∩ D ,k − = ∅ , then the same argument as above works.If D j + ,k − ∩ D ,j + = ∅ or D j + ,k − ∩ D ,k − = ∅ , we interchange the roles of x and x and get the equality | b j + | = | b j + | + | a j + ( b j + + b k − ) | , hence b j + + b k − = 0 . Thus again weobtain (6.5).
AXIMALLY REDUCIBLE MONODROMY OF BIVARIATE HYPERGEOMETRIC SYSTEMS 33
If we recall the condition 1 + r P j =1 a j = 1 + r P j =1 b j = 0 , m = r + 2 then we see that thematrix A ′ with maximally reduced monodromy Horn( A ′ , c ′ ) must be either(6.7) − − a b − a − b ... ... a r/ − b r/ − − a r/ − − b r/ − , r : even . or(6.8) − − a b − a − b ... ... a ( r − / b ( r − / − a ( r − / − b ( r − / , r : odd . Elementary plane geometry shows that the matrix A ′ like (6.7) produces a zonotopeOre-Sato polygon.To examine the case (6.8) we shall use the notation A − = ( − , − , − ∈ I − . For j + ∈ I + we see that either D j + , − ∩ D ,j + = ∅ or D j + , − ∩ D , − = ∅ holds.If D j + , − ∩ D ,j + = ∅ the analytic continuation of the solution u ,j + ( x ) = X n,m ≥ Res α j + ( s ) + γ j + = − m,s + δ = − n ϕ , − ,j + ( s ) x s , to u j + , − ( x ) = X n,m ≥ Res α j + ( s ) + γ j + = − m, − s − s + γ − = − n ϕ , − ,j + ( s ) x s , by Mellin-Barnes contour throw on the complex plane { s ∈ C : α j + ( s ) + γ j + = − m, m ∈ Z ≥ } . The argument using Theorem 3.7 (2) would entail the equality a j + = a j + + | a j + − b j + | . This means that a j + − b j + = 0 . If D j + , − ∩ D j + , − = ∅ , the same argument on the analytic continuation u , − ( x ) → u j + , − ( x ) yields the equality 1 = 1 + | a j + − b j + | . Hence we get a j + − b j + = 0 again i.e. A j + is collinear to ( − , − . (See Fig. 5.)In an analogous way we can examine the analytic continuation of u , − ( x ) = X n,m ≥ Res − s − s + γ − = − m,s + δ = − n ϕ , − ,j + ( s ) x s , C j + , − C ∨ j + , − C ∨ , − C , − C ,j + C ∨ ,j + Figure 5.
Recession cones intersectionto u ,k − ( x ) = X n,m ≥ Res α k − ( s ) + γ k − = − m,s + δ = − n ϕ , − ,j + ( s ) x s , by Mellin-Barnes contour throw along the complex planes { s ∈ C : s + δ ∈ Z ≤ } . In view of the relation C ∨ , ⊂ C ∨ ,k − , we see that 1 + | a k − | = 1 i.e. | a k − | = 0 and A k − is collinear to (0 , . We can now apply the same argument to the residues of ϕ , − ,j + ( s ) x s and ϕ , − ,k − ( s ) x s . In this way we can conclude that every row vector of the matrix (6.8) is collinear toone of three vectors (1 , , (0 , , ( − , − . This means that the Ore-Sato polygon of the Horn system Horn( A ′ , c ′ ) with A ′ of (6.8)must be a Minkowski sum of a triangle and segments parallel to the sides of it. (cid:3) Corollary 6.7.
A bivariate hypergeometric system Horn(
A, c ) has a maximally reduciblemonodromy representation if and only if the solution space of Horn( A, ˜ c ) is spanned byPuiseux polynomials for some choice of the vector of parameters ˜ c .Proof. If the solution space of the system Horn( A, ˜ c ) is spanned by Puiseux polynomials,evidently its monodromy is maximally reducible.Proposition 6.6 shows that the Ore-Sato polygon of a hypergeometric system Horn( A, c )with a maximally reducible monodromy must be either a zonotope or the Minkowski sumof a triangle and segments parallel to its sides. After Proposition 6.5, Horn( A, ˜ c ) admitsa Puiseux polynomial basis for a suitably chosen parameter ˜ c. (cid:3) AXIMALLY REDUCIBLE MONODROMY OF BIVARIATE HYPERGEOMETRIC SYSTEMS 35
Example 6.8.
A random zonotope.
Let us consider the following configuration which isgiven by the Minkowski sum of four segments:(6.9) A = − − − − − −
23 22 − − . ✻ ✲(cid:0)✁✁❏❏❏❏❍❍(cid:0)✁✁❏❏❏❏❍❍ rr r r rr r r rr r r r r rr r r r r rr r r rr r r rr Figure 6.
The zonotope which defines the matrix (6.9)
Choose the vector of parameters to be c = (3 , − , − , , − , − , − , − . The correspond-ing hypergeometric system Horn(
A, c ) is holonomic with rank 31. Here is the pure Puiseuxpolynomial basis in its solution space (which was computed with Mathematica 9.0). Thepersistent solutions are x , x x , √ x x / , x x , x / x / , x x while non-persistent Puiseux polynomialsolutions are x x , x / x / , x / x / , x / x / , √ x x / , x / x / , x x , x x + 18900 x x + 74529 x x + 715715 x x , x /
71 7 √ x + x / √ x , x / x / − x / x / , x / √ x − √ x x / , x − , √ x x / − √ x x / , x / x / − √ x x / , x / x / − x / x / , x / √ x − x / x / , x x − x x , x / x / − x / x / , x /
31 3 √ x − x / x / , x x / + x x / − x x / , − x / x / + x / x / + x / x / , x x − x x , x / x / + x / x / , x / x / + √ x x / , x / x / − x / x / . The following picture depicts the supports of the above solutions to Horn(
A, c ) . The bigbullets correspond to monomials (both persistent and not) while the small bullets to allother solutions. The parallelograms that carry the supports arise as intersections of thedivisors of the defining Ore-Sato coefficient. - - - Figure 7.
The supports of the solutions to Horn(
A, c ) defined by (6.9)
Example 6.9.
The sum of a triangle and its sides.
Let us consider the following config-uration which is given by the Minkowski sum of a triangle and all of its sides:(6.10) A = − − − − − −
31 2 − − − − . Choose the vector of parameters to be c = ( − , − , , − , − , , , − , − . The corre-sponding hypergeometric system is holonomic with rank 40 and is defined by the followingdifferential operators: x ( θ − θ + 5)(2 θ − θ − θ − θ − θ − θ − θ − θ )( θ + 2 θ + 3) − ( θ + 2 θ + 6)( θ + 2 θ + 1)(2 θ − θ − θ − θ − θ − θ + 10)( θ − θ + 2) , AXIMALLY REDUCIBLE MONODROMY OF BIVARIATE HYPERGEOMETRIC SYSTEMS 37 ✻ ✲✏✏✏✁✁✁✁❍❍✏✏✏✏✏✏✁✁❍❍❍❍ rr r r rr r r rr r r r r rr r r r r rrr r rrrr rrrrrr rrr r = ✻ ✲✁✁✏✏✏❍❍ r r rr r + ✻ ✲❍❍ r r + ✻✲✁✁ r r + ✻ ✲✏✏✏ r r Figure 8.
The polygon defining the matrix (6.10) and its Minkowski decomposition x ( θ − θ )( θ − θ + 1)( θ − θ + 2)( θ − θ + 8)( θ − θ + 9) ( θ − θ + 10)(2 θ − θ − θ +2 θ +3)( θ +2 θ +4) − ( θ − θ +5)( θ − θ +6)( θ − θ +7)(2 θ − θ − θ − θ − θ + 2 θ )( θ + 2 θ + 1)( θ + 2 θ + 5)( θ + 2 θ + 6) . This system has the following five persistent Puiseux polynomial solutions (which actu-ally turn out to be monomials): x x , x x , x / x / , x / x / , x / x / . The followingthirty pure Puiseux polynomial solutions to Horn(
A, c ) were computed with Mathemat-ica 9.0:
28 + 15 /x , x − / x − / (7 x + 22 x + 44 x x ) , x − / x − / (196 + 297 x + 231 x x ) ,x − / x − / (198 + 140 x + 165 x x ) , x − / x / (25 + 120 x + 72 x ) ,x / x − / (3 + 1254 x + 52 x x ) ,x / x / (298452 + 129675 x + 27930 x x + 588 x x + 85 x x ) ,x / x − / (91 + 15 x + 15675 x + 3135 x x ) , x − (1040 + 819 x + 62700 x x ) ,x / x − / (2340 + 182 x + 72675 x ) , x / x / (8892 + 266 x + 105 x + 72 x x ) ,x x (426360 + 34884 x + 26600 x x + 1200 x x + 51 x x ) ,x / x / (43605 + 741 x + 3325 x + 1125 x x ) ,x / x / (46512 + 6669 x + 900 x x + 64 x x ) , x + 34884 x + 51 x + 4500 x x + 74100 x x /x ,x − / x − / (8151 x + 9 x + 1980 x x + 73150 x x + 639540 x x ) ,x − / x / (1200 + 33345 x + 170544 x + 336 x + 13300 x x ) ,x − / x / (32 + 1596 x + 17442 x + 38760 x + 105 x x ) ,x − / x / (17 + 1575 x + 31122 x + 149226 x ) ,x / x − / (16 x + 48279 x + 18018 x x ) ,x / x − / (33 x + 9996 x + 3672 x x + 22100 x x + 1326 x x + 4641 x x + 2652 x x ) ,x / x − / (81 + 3024 x + 192 x x + 5720 x + 1872 x x + 624 x x + 72 x x ) ,x x − (420 + 216 x + 2925 x x + 175 x x + 2145 x x + 819 x x ) ,x / x − / (23520 + 1728 x + 109200 x + 34125 x x + 38220 x x + 2912 x x ) ,x / x − / (9504 + 990 x + 128700 x + 41580 x x + 113256 x x + 7280 x x + 4455 x x ) ,x − / x − / (1225 x + 3780 x + 1512 x + 75 x + 2730 x x + 18018 x x + 27300 x x ) ,x − x − (120 x + 216 x + 45 x + 819 x x + 3250 x x + 2925 x x ) ,x − / x − / (3456 x + 2835 x + 5824 x + 65520 x x + 163800 x x + 82320 x x +38220 x x ) , x − / x − / (66 x + 2652 x x + 12852 x x + 11424 x x +1377 x + 18564 x x + 48620 x x ) , x − / x − / (198 x + 1456 x + 10725 x x +16632 x x + 3696 x x + 3432 x + 18876 x x ) . We omit the remaining five solutions since they are too cumbersome to display. Theirinitial exponents are ( − / , / , ( − / , / , ( − / , / , ( − / , / , ( − , . References [1] F. Beukers.
Algebraic A-hypergeometric functions , Invent. Math. , no. 3 (2010), 589-610.[2] F. Beukers.
Monodromy of A-hypergeometric functions , arXiv.org 1101.0493v2 (2013), 27pp.[3] F. Beukers and G. Heckman.
Monodromy for the hypergeometric function n F n − , Invent.Math. , no. 2 (1989), 325-354.[4] E. Cattani, A. Dickenstein, and B. Sturmfels. Rational hypergeometric functions , CompositioMath. (2001), 217-240.[5] A. Dickenstein, L. Matusevich, and T. Sadykov.
Bivariate hypergeometric D-modules , Adv.in Math. , no. 1 (2005), 78-123.[6] A. Dickenstein, L. Matusevich, and E. Miller.
Binomial D-modules , Duke Math.J. , no. 3(2010), 385-429.[7] B. Feng, Y.-H. He, K.D. Kennaway, and C. Vafa.
Dimer models from mirror symmetry andquivering amoebae , Adv. Theor. Math. Phys. , no. 3 (2008), 489-545.[8] I.M. Gelfand, M.I. Graev, and V.S. Retach. General hypergeometric systems of equations andseries of hypergeometric type , Russian Math. Surveys , no. 4 (1992), 1-88.[9] I.M. Gelfand, M.M. Kapranov, and A.V. Zelevinsky. Hypergeometric functions and toric va-rieties , Funktsional. Anal. i Prilozhen. , no. 2 (1989), 12-26.[10] R.M. Hain and R. MacPherson. Higher logarithms , Illinois Journal of Math. , no. 2 (1990),392-475.[11] V.P. Palamodov. Linear Differential Operators with Constant Coefficients (Russian), Nauka,1967.[12] M. Passare, T.M. Sadykov, and A.K. Tsikh.
Nonconfluent hypergeometric functions in sev-eral variables and their singularities , Compos. Math. 141, no. 3 (2005), 787-810.[13] M. Passare and A. Tsikh.
Amoebas: their spines and their contours , Idempotent Math-ematics and Mathematical Physics : International workshop, February 3-10, 2003, Er-win Schr¨odinger International Institute for Mathematical Physics, Vienna, Austria / Eds.G.L. Litvinov, V.P. Maslov. AMS, 2005. V. 377.[14] T.M. Sadykov.
On a multidimensional system of hypergeometric differential equations ,Siberian Math. J. (1998), 986-997.[15] T.M. Sadykov. On the Horn system of partial differential equations and series of hypergeo-metric type , Math. Scand. (2002), 127-149.[16] T.M. Sadykov. The Hadamard product of hypergeometric series , Bull. Sci. Math. , no. 1(2002), 31-43.[17] T.M. Sadykov.
Hypergeometric systems of equations with maximally reducible monodromy ,Doklady Math. , no. 3 (2008), 880-882.[18] T.M. Sadykov. Hypergeometric systems with polynomial bases , Journal of Siberian FederalUniversity. Mathematics & Physics , (2008), 25-32.[19] M. Sato. Theory of prehomogeneous vector spaces (algebraic part) , Nagoya Math. J. ,(1990), 1-34.[20] S. Tanab´e.
On Horn-Kapranov uniformisation of the discriminantal loci , Advanced Studiesin Pure Mathematics. , (2007), 223-249. AXIMALLY REDUCIBLE MONODROMY OF BIVARIATE HYPERGEOMETRIC SYSTEMS 39
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