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MONODROMY OF HOLONOMIC HYPERGEOMETRIC SYSTEMS Research statement by T.M. Sadykov A linear homogeneous holonomic system of partial differential equations with constant coefficients and one unknown function depending on n variables (z1 , . . . , zn ) Cn has the form Qi ,..., z1 z f (z1 , . . . , zn ) = 0,
n

i = 1, . . . , n.

(1)

Here Q1 , . . . , Qn are polynomials with a finite number of common zeros in Cn . The solutions of (1) are described by the Palamodov-Malgrange-Ehrenpreis fundamental principle (see [14], Chapter 6). The class of hypergeometric systems of partial differential equations includes systems of the form xi Pi ()y (x) = Qi ()y (x), i = 1, ..., n. (2) Here Pi , Qi are polynomials in n variables, = (1 , ..., n ), and i = xi i . Hypergeometric x systems constitute an important class of regular holonomic ideals in the Weyl algebra of partial differential operators with polynomial coefficients. They include the Gelfand-KapranovZelevinsky system, the Knizhnik-Zamolodchikov equations as well as the Horn and Mellin systems of partial differential equations. It is easy to see that (1) can be treated as a special instance of a hypergeometric system of partial differential equations. Indeed, introducing new variables xi = ezi , i = 1, . . . , n and denoting y (x1 , . . . , xn ) = f (log x1 , . . . , log xn ), we write the system (1) in the form Qi (x1 , . . . , xn )y (x) = 0, i = 1, . . . , n,

where xi = xi i . Thus a general linear homogeneous holonomic system of partial differential x equations with constant coefficients can be viewed as a special instance of the hypergeometric system (2) corresponding to the case P1 . . . Pn 0. On the other hand, the system (2) can be considered as a linear perturbation of an ideal in the commutative subalgebra C[x1 , . . . , xn ] in the n-variate Weyl algebra Dn . Thus the class of hypergeometric systems can be viewed as the simplest class of systems of partial differential equations with nonconstant polynomial coefficients. In contrast with the case of constant coefficients, little is known about the global properties of a general hypergeometric system of equations. The monodromy of a multi-valued analytic function characterizes the branching of this function on its singular locus. This notion gave rise to a number of classical problems (including Hilbert's 21st problem) and is at the core of several open conjectures. The main problem in the proposed research pro ject is to compute the monodromy group of the general hypergeometric system of partial differential equations. The modern approach to the study of monodromy of hypergeometric differential equations was pioneered by P. Deligne in his seminal papers on hypergeometric functions and non-lattice 1


integral monodromy (see [3]). One of unsolved monodromy-related problems is the CattaniDickenstein-Sturmfels conjecture (see [2]) stating that the class of GKZ-rational configurations coincides with the class of essential Cayley configurations. In order to formulate the expected results of the research we need to introduce the main ob jects of study in the proposed pro ject. In order to accommodate two different sets of variables, we denote by Dn the Weyl algebra with generators x1 , . . . , xn , x1 , . . . , xn , and by Dm the Weyl algebra whose generators are y1 , . . . , ym , y1 , . . . , ym . We set xj = xj xj for 1 j n, and yi = yi yi , for 1 i m. We also define x = (x1 , . . . , xn ) and y = (y1 , . . . , ym ). When the meaning is clear, we will drop many of the subindices to simplify the notation. We fix a matrix A = (aij ) Z(n-m)вn of full rank n - m whose first row is the vector (1, . . . , 1), and a matrix B Znвm = (bj i ) of full rank m such that A · B = 0. For 1 j m, set bj = (bj 1 , . . . , bj m ) Zm the j -th row of B . The (positive) greatest common divisor of the maximal minors of the matrix B is denoted by g . For i = 1, . . . , m, and a fixed parameter vector c = (c1 , . . . , cn ) Cn , we let
|bj i |-1

Pi =
bj i <0 l=0 bj i -1

(bj · y + cj - l),

(3)

Qi =
bj i >0 l=0

(bj · y + cj - l), and

(4) (5)

Hi = Qi - yi Pi ,

where bj · y = m bj k yk . The operators Hi are the Horn operators corresponding to the k=1 lattice LB = {B · z : z Zm } and the parameter vector c. We call di = bij >0 bij = - bij <0 bij the order of the operator Hi . Definition 1. The Horn system is the following left ideal of Dm : Horn (B , c) = H1 , . . . , H
m

Dm .

Now denote by b(i) the columns of the matrix B . Any vector u Rn can be written as u = u+ - u- , where (u+ )i = max(ui , 0), and (u- )i = - min(ui , 0). For i = 1, . . . , m, we let: Ti = x+ - x- ,
v v1 vn here we use multi-index notation x = x1 · · · xn . More generally, for any u LB , set u u Tu = x + - x - . b
(i)

b

(i)

These are the lattice operators arising from LB . Definition 2. The lattice ideal arising from LB is: IB = Tu : u L
B

C[x1 , . . . , xn ]. 2


Recall that the toric ideal corresponding to A is: IA = Tu : u ker Z (A) C[x1 , . . . , xn ]. We will also denote: I = T1 , . . . , T
m

C[x1 , . . . , xn ].

The ideal I is called a lattice basis ideal. Notice that, for m = 2, I is a complete intersection. This is not necessarily true if m > 2. There is a natural system of differential equations arising from a toric ideal IA and a parameter vector. This system, called the A-hypergeometric system with parameter A · c, is defined as: n HA (A · c) = IA +
j =1 n From now on we will use the notation A · - A · c to mean j =1 aij xj xj - (A · c)i : i = 1, . . . , n - m . A-hypergeometric systems were first defined by Gelfand, Graev and Zelevinsky in [7], and their systematic analysis was started by Gelfand, Kapranov and Zelevinsky (see, for instance, [8]). Saito, Sturmfels and Takayama have used GrЁbner deformations in the Weyl o algebra to study A-hypergeometric systems (see [15]). There is a well-known link between the properties of solutions of a regular holonomic ideal and the monodromy of the corresponding system of equations. Trivial monodromy of a system of differential equations means that the system admits a basis of meromorphic solutions; finite monodromy corresponds to algebraic solutions while reducible monodromy of an ordinary differential operator means that it can be factorized. The monodromy of the ordinary hypergeometric differential equation of arbitrary order was described by Beukers and Heckman in [1] in terms of Shephard-Todd groups. The monodromy of special hypergeometric ideals generated by operators of low order in two or three variables was calculated in [9, 10]. Monodromy of hypergeometric ideals with algebraic solutions have been studied in [11] and [16]. Various properties of the monodromy of A-hypergeometric systems have been recently considered in [17]. However, despite its importance, the problem of computing the monodromy group of a general hypergeometric system of partial differential equations is still open.

aij xj xj - (A · c)i : i = 1, . . . , n - m Dn .

The exp ected results of the research are as follows: Theorem 3. Any series solution (centered at the origin) of a Horn system with generic parameters is either a ful ly supported series or a stable Puiseux polynomial. Theorem 4. The set of supports of solutions to a hypergeometric system with generic parameters consists of supports of solutions to associated atomic systems. In particular, the initial exponents of Puiseux polynomial solutions to a hypergeometric system are precisely the initial exponents of Puiseux polynomials which satisfy the associated atomic systems.

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Theorem 5. Any Horn system defined by a matrix whose rows are the vertices of a simplex or a paral lelepiped admits a basis of Puiseux polynomials for suitable values of its parameters. In particular, the solution space of such a system splits into the direct sum of one-dimensional invariant subspaces. The monodromy representation of such a Horn system is reducible. Theorem 6. In the case of two variables, suppose that (Mj , Mk ) = 0,
Mj ,M
k

lin. indep.

where the summation is over al l pairs of linearly independent rows of the matrix defining the Horn system. Then for generic parameter vector monodromy representations of the Horn systems Horn(A, ) and Horn(A, - ej ) are equivalent for any j = 1, . . . , m. We also expect theorems giving necessary and/or sufficient conditions for the monodromy of a general hypergeometric system to be trivial, finite or reducible. Another ob jective is to obtain an explicit decomposition of the solution space of a regular holonomic hypergeometric ideal into invariant (under the action of monodromy) subspaces and to describe hypergeometric ideals with irreducible monodromy. In particular, describing the class of hypergeometric ideals with trivial monodromy would prove the Cattani-Dickenstein-Sturmfels conjecture. Several results related to monodromy of hypergeometric systems have been recently obtained by my co-authors and myself. These results include the following theorems (see [4],[6] and [13]). Theorem 7. The number of subspaces in the solution space of a hypergeometric system that are invariant under the action of monodromy is bounded from below by the defect of the corresponding toric ideal. The latter can be explicitly computed in combinatorial terms. Theorem 8. A bivariate Horn system is generical ly holonomic. Definition 9. In the case that m = 2, we set ij = min(|bi1 bj 2 |, |bj 1 bi2 |), 0 if bi , bj are in the interior of opposite quadrants of Z2 , otherwise,

for 1 i, j n. The number ij is called the index associated to bi and bj . Theorem 10. Let B be an n в 2 integer matrix of ful l rank such that its rows b1 , . . . , bn satisfy b1 + · · · + bn = 0. If c Cn is a generic parameter vector, then the ideals Horn (B , c) and HB (c) are holonomic. Moreover, rank (HB (c)) = rank (Horn (B , c)) = d1 d2 - bi , bj
dependent

ij = g · vol (A) + bi , b
j
independent

ij ,

where the first summation runs over linearly dependent pairs bi , bj of rows of B that lie in opposite open quadrants of Z2 , and the second summation runs over linearly independent such pairs. 4


Theorem 11. The monodromy of the Mel lin system is always reducible. The dimension of the space of its algebraic solutions can be explicitly computed in combinatorial terms. Let S P be a finite set with m+1 elements, E be a finite set, and Fm 1 (P\S ) Aut(E ) = be a group homomorphism into the group of permutations of E . Here Fm denotes the free group on m generators. Assume the image acts transitively on E . Can we realize the elements of E as distinct holomorphic function germs at a point in P \ S , such that E spans the solution space of a homogeneous Fuchsian equation whose singularities lie in S and the group homomorphism is the monodromy given by analytic continuation around the points of S ? We call this question the discrete Riemann-Hilbert problem associated to the given data. There is a profound link between the monodromy of a hypergeometric function and the dessin d'enfant (see [12]) defined by the Riemann surface of that function. Dessins d'enfants correspond bijectively to the data of two permutations acting transitively on a finite set, and to finite branched covering spaces of P, branched over 0, 1, and (up to isomorphisms). Such covering maps are called Belyi maps. The dessin sits in the covering space as the preimage of the segment [0, 1]. Its complement is a union of cells, or faces, each containing one preimage of . If the dessin has v vertices, e edges, and f faces, then v - e + f equals the Euler characteristic of the covering space. A plane tree is a tree with a cyclic ordering of the edges at each vertex. There is a bijective correspondence between equivalence classes of Shabat polynomials and isomorphism classes of plane trees. (Here, we do not distinguish the two critical points of a Shabat polynomial or the two bicolourings of a tree.) An n-dimensional linear representation of a dessin realizes its edges as distinct vectors in an n-dimensional complex vector space and the permutations of the dessin as linear automorphisms. Every dessin has a linear representation by permutation matrices in a space of dimension equal to its number of edges. It is easy to see that the plane trees that have a one-dimensional linear representation are precisely the stars. Theorem 12. Let T be a plane tree. The fol lowing are equivalent. (i) T has a linear representation of dimension at most two. (ii) The Riemann-Hilbert problem for T has a solution of order at most two. (iii) The Riemann-Hilbert problem for T has a hypergeometric solution of order at most two. (iv) T is a star, a 2-star, or a chain. As for my teaching experience, it is comprised of the courses that I taught at the University of Western Ontario (London, Canada), Stockholm University (Sweden) and Siberian Federal University (Krasnoyarsk, Russia). I have been teaching regularly on both undergraduate and graduate level. The courses that I have taught include Calculus courses on different levels, an advanced course in Topology for undergraduate and graduate students, full-year courses in Functional Analysis, The Theory of Functions of a Real Variable, The Theory of Functions of a Complex Variable, Computer Algebra as well as graduate courses in The Theory of Functions of Several Complex Variables and Homology.

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References
[1] F. Beukers and G. Heckman. Monodromy for the hypergeometric function n F (1989), 325-354.
n-1

, Invent. Math. 95

[2] E. Cattani, A. Dickenstein and B. Sturmfels, Rational hypergeometric functions, Compositio Math. 128 (2001), 217-240. [3] P. Deligne and G.D. Mostow. Monodromy of hypergeometric functions and non-lattice integral monodromy. Publications Mathematiques de l'IHS 63 (1986), 5-89. [4] A. Dickenstein, L. Matusevich and T. Sadykov. Bivariate hypergeometric D-modules. Adv. in Math., 196, no. 1 (2005), 78-123. [5] A. Dickenstein and T. Sadykov. Algebraicity of solutions to the Mel lin system and its monodromy. Doklady of Russian Academy of Sciences 412, (2007), no. 4, 1 - 3. [6] A. Dickenstein and T. Sadykov. Bases in the solution space of the Mel lin system. Sbornik Mathematics 198, (2007), no. 9, 59 - 80. [7] I. M. Gelfand, M. I. Graev, and A. V. Zelevinsky. Holonomic systems of equations and series of hypergeometric type. Dokl. Akad. Nauk SSSR, 295(1):14­19, 1987. [8] I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky. Hypergeometric functions and toric varieties. Funktsional. Anal. i Prilozhen., 23(2):12­26, 1989. [9] M. Kato. Appel l's F4 with finite irreducible monodromy group, Kyushu Journal of Mathematics 51:1 (1997). [10] M. Kato. Appel l's Hypergeometric Systems F2 with Finite Irreducible Monodromy Groups, Kyushu Journal of Mathematics 54:2 (2000). [11] M. Kato and M. Noumi. Monodromy groups of hypergeometric functions satisfying algebraic equations, Tohoku Math. J. 55 (2003), 189-205. [12] S. K. Lando and A. K. Zvonkin. Graphs on surfaces and their applications. With an appendix by Don B. Zagier. Encyclopaedia of Mathematical Sciences 141. Low-Dimensional Topology II. Springer-Verlag, 2004. [13] F. Larusson and T. Sadykov. Dessins d'enfants and differential equations. Algebra and Analysis 19 (2007), no. 6, 16 pages. [14] V.P. Palamodov. Linear Differential Operators with Constant Coefficients (Russian), Nauka, 1967. [15] Mutsumi Saito, Bernd Sturmfels, and Nobuki Takayama. GrЁ er Deformations of Hypergeoobn metric Differential Equations. Springer-Verlag, Berlin, 2000. [16] S. Tanabe. Logarithmic vector fields and multiplication table, math.AG/0602301. [17] U. Walther. Duality and arXiv:math.AG/0508622. monodromy reducibility of A-hypergeometric systems,

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