%0 Journal Article
%T On the rank of the holomorphic solutions of PDE associated to directed graphs
%J AUT Journal of Mathematics and Computing
%I Amirkabir University of Technology
%Z 2783-2449
%A Damadi, Hamid
%A Rahmati, Farhad
%D 2021
%\ 02/01/2021
%V 2
%N 1
%P 1-9
%! On the rank of the holomorphic solutions of PDE associated to directed graphs
%K Directed graph
%K Binomial $D$-module
%K Lattice basis ideal
%R 10.22060/ajmc.2020.18413.1031
%X 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.
%U https://ajmc.aut.ac.ir/article_4121_c864d5914fe43e4c473524a899cf8c6a.pdf