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On the rank of the holomorphic solutions of PDE associated to directed graphs | ||
| AUT Journal of Mathematics and Computing | ||
| مقاله 1، دوره 2، شماره 1، اردیبهشت 2021، صفحه 1-9 اصل مقاله (397.31 K) | ||
| نوع مقاله: Original Article | ||
| شناسه دیجیتال (DOI): 10.22060/ajmc.2020.18413.1031 | ||
| نویسندگان | ||
| Hamid Damadi* ؛ Farhad Rahmati | ||
| Department of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran | ||
| چکیده | ||
| 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. | ||
| کلیدواژهها | ||
| Directed graph؛ Binomial $D$-module؛ Lattice basis ideal | ||
| مراجع | ||
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