Amirkabir University of TechnologyAUT Journal of Mathematics and Computing2120210201On the rank of the holomorphic solutions of PDE associated to directed graphs19412110.22060/ajmc.2020.18413.1031ENHamidDamadiDepartment of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), Tehran, IranFarhadRahmatiDepartment of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), Tehran, IranJournal Article20200511Let G be a directed graph with m vertices and n edges, I(B) the<br />binomial ideal associated to the incidence matrix B of the graph G, and I_L the lattice<br />ideal associated to the columns of the matrix B. Also let B_i be a submatrix of<br />B after removing the ith column. In this paper it is determined that which prime<br />minimal ideals of I(B_i) are Andean or toral. Then we study the rank of the space<br />of solutions of binomial D-module associated to I(B_i) as A-graded ideal, where A is<br />a matrix that, AB_i = 0. Afterwards, we define a maximal cellular cycle and prove<br />that for computing this rank it is enough to consider these components of G. We<br />introduce some bounds for the number of the vertices of the convex hull generated<br />by the columns of the matrix A. Finally an algorthim is introduced by which we can<br />compute the volume of the convex hull corresponded to a cycles with k diagonals, so<br />by Theorem 2.1 the rank of (D / H_A(I(B_i); beta)) can be computed.https://ajmc.aut.ac.ir/article_4121_93e1173b7519f0c5363021779bf5afe9.pdf