Document Type : Original Article

**Authors**

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

**Abstract**

Let $G$ be a directed graph with $m$ vertices and $n$ edges, $I(\textbf{B})$ the binomial ideal associated to the incidence matrix $\textbf{B}$ of the graph $G$, and $I_L$ the lattice ideal associated to the columns of the matrix $\textbf{B}$. Also let $\textbf{B}_i$ be a submatrix of $\textbf{B}$ after removing the $i$th column. In this paper it is determined that which minimal prime ideals of $I(\textbf{B}_i)$ are Andean or toral. Then we study the rank of the space of solutions of binomial $D$-module associated to $I(\textbf{B}_i)$ as $\textbf{A}$-graded ideal, where $\textbf{A}$ is a matrix that, $\textbf{A}\textbf{B}_i=0$. 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 $\textbf{A}$. 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 $\frac{D}{H_{\textbf{A}}(I(\textbf{B}_i), \boldsymbol{\beta})}$ can be computed.

**Keywords**

**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.

February 2021

Pages 1-9