On the rank of the holomorphic solutions of PDE associated to directed graphs

Document Type : Original Article


Department of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran



Let G be a directed graph with m vertices and n edges, I(B) the binomial ideal associated to the incidence matrix B of the graph G, and IL the lattice ideal associated to the columns of the matrix B. Also let Bi be a submatrix of B after removing the ith column. In this paper it is determined that which minimal prime ideals of I(Bi) are Andean or toral. Then we study the rank of the space of solutions of binomial D-module associated to I(Bi) as A-graded ideal, where A is a matrix that, ABi = 0. Afterwards, we define a miniaml cellular cycle and prove that for computing this rank it is enough to consider these components of G. We introduce some bounds for the number of the vertices of the convex hull generated by the columns of the matrix A. Finally an algorthim is introduced by which we can compute the volume of the convex hull corresponded to a cycles with k diagonals, so by Theorem 2.1 the rank of (D / H_A(I(B_i); beta)) can be computed.


Main Subjects

[1] V. Batyrev, Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J. Algebraic Geom., 3(3) (1994) 493-535.
[2] B. Braun, An Ehrhart series formula for reflexive polytopes, Electron. J. Combin. 13 (1) (2006), Note 15, 5 pp. (electronic).
[3] A. Dickenstein, L. Matusevich, E. Miller, Combinatorics of binomial primary decomposition, Math. Z. 264 (4) (2010) 745-763.
[4] A. Dickenstein, L. Matusevich, E. Miller, Binomial D-modules, Duke Math. J. 151 (3) (2010) 385-429.
[5] D. Eisenbud, B. Sturmfels, Binomial ideals, Duke Math. J. 84 (1) (1996), 1-45.
[6] E. Ehrhart, Polynˆomes Arithm´etiques et M´ethode des Poly`edres en Combinatoire, Birkhuser, Boston, Basel, Stuttgart, 1977.
[7] K. Fischer, J. Shapiro, Mixed matrices and binomial ideals, J. Pure Appl. Algebra 113 (1) (1996) 39-54. [8] I. M. Gelfand, M. I. Graev, A. V. Zelevinski, Holonomic systems of equations and series of hypergeometric type, Dokl. Akad. Nauk SSSR, 295 (1) (1987) 14-19.
[9] G. Hegedus, A. M. Kasprzyk, The boundary volume of a lattice polytope, Bull. Aust. Math. Soc. 85 (2012), 84-104.
[10] T. Hibi, Dual polytopes of rational convex polytopes, Combinatorica, 12(2) (1992) 237-240.
[11] S. Hos¸ten, J. Shapiro, Primary decomposition of lattice basis ideals,J. Symbolic Comput. 29 (2000), no. 4-5, 625-639.