On matrix and lattice ideals of digraphs

Document Type : Research Paper


1 Department of Mathematics, Amirkabir University of Technology (Tehran Polytechnic) Tehran, Iran.

2 Amirkabir University of Technology


‎Let $\textit{G}$ be a simple‎, ‎oriented connected graph with $n$ vertices and $m$ edges‎. ‎Let $I(\textbf{B})$ be the binomial ideal associated to the incidence matrix \textbf{B} of the graph $G$‎. ‎Assume that $I_L$ is the lattice ideal associated to the rows of the matrix $\textbf{B}$‎. ‎Also let $\textbf{B}_i$ be a submatrix of $\textbf{B}$ after removing the $i$-th row‎. ‎We introduce a graph theoretical criterion for $G$ which is a sufficient and necessary condition for $I(\textbf{B})=I(\textbf{B}_i)$ and $I(\textbf{B}_i)=I_L$‎. ‎After that we introduce another graph theoretical criterion for $G$ which is a sufficient and necessary condition for $I(\textbf{B})=I_L$‎. ‎It is shown that the heights of $I(\textbf{B})$ and $I(\textbf{B}_i)$ are equal to $n-1$ and the dimensions of $I(\textbf{B})$ and $I(\textbf{B}_i)$ are equal to $m-n+1$; then $I(\textbf{B}_i)$ is a complete intersection ideal‎.


Main Subjects

[1] C. Berkesch-Zamaere, L. Matusevich and U. Walther, Torus equivariant Dmodules and hypergeometric systems, Preprint, arXiv (2013): 1308:5901.
[2] A. Dickenstein, L. Matusevich and E. Miller, Combinatorics of binomial primary decomposition, Math. Z., 264 (2010) 745–763.
[3] A. Dickenstein, L. Matusevich and E. Miller, Binomial D-modules, Duke Math. J., 151 (2010) 385–429.
[4] K. Eto, Cohen-Macaulay rings associated with oriented digraphs, J. Algebra, 206 (1998) 541–554.
[5] K. Eto, A free resolution of a binomial ideal, Comm. Algebra, 27 (1999) 3459–3472.
[6] D. Eisenbud and B. Sturmfels, Binomial ideals, Duke Math. J., 84 (1996) 1–45.
[7] K. Fischer and J. Shapiro, Mixed matrices and binomial ideals, J. Pure and Appl. Algebra, 113 (1996) 39–54.
[8] S. Hosten and J. Shapiro, Primary decomposition of lattice basis ideals, J. Symbolic Comput, 29 (2000) 625–639.
[9] H. Matsumura, Commutative algebra, Second edition, Mathematics Lecture Note Series, 56, Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980.
[10] A. Schrijver, Theory of Linear and Integer Programming, A Wiley-Interscience Publication. John Wiley Sons, Ltd., Chichester, 1986.
  • Receive Date: 02 August 2017
  • Revise Date: 28 October 2017
  • Accept Date: 18 November 2017
  • Published Online: 01 June 2018