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 I_L the lattice
ideal associated to the columns of the matrix B. Also let B_i be a submatrix of
B after removing the ith column. In this paper it is determined that which prime
minimal ideals of I(B_i) are Andean or toral. Then we study the rank of the space
of solutions of binomial D-module associated to I(B_i) as A-graded ideal, where A is
a matrix that, AB_i = 0. Afterwards, we define a maximal 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.


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