arXiv subjects
Publications and source records attributed to Humberto Muñoz-George.
Inspired by the Erdős Rényi model, we propose a new model for freesquare random monomial ideals generated by edges and covers of a graph. This permit us to investigate the conditions of normality for which we obtain asymptotic results. We also elaborate on asymptotic results for other invariants such as the Krull dimension (for which we obtain threshold function), the regularity and the $v$-number.
Let $G$ be a graph and let $J=I_c(G)$ be its ideal of covers. The aims of this work are to study the {\rm v}-number ${\rm v}(J)$ of $J$ and to study when $J$ is linearly presented using combinatorics and commutative algebra. We classify when ${\rm v}(J)$ attains its minimum and maximum possible values in terms of the vertex covers of the graph that satisfy the exchange property. If the cover ideal of a graph has a linear presentation, we express its v-number in terms of the covering number of the graph. If $G$ is unmixed, the graph $\mathcal{G}_J$ of $J$ is the graph whose vertices are the minimal vertex covers of $G$ and whose edges are the pairs $\{C,C'\}$ such that $|C\cup C'|=|C|+1$. We show necessary and sufficient conditions for the graph $\mathcal{G}_J$ of $J$ to be connected. Then, for unmixed König graphs, we classify when $J$ is linearly presented using graph theory, and show some results on Cohen--Macaulay König graphs. If $G$ is unmixed, it is shown that the columns of the linear syzygy matrix of $J$ are linearly independent if and only if $\mathcal{G}_J$ has no strong $3$-cycles. One of our main theorems shows that if $G$ is unmixed and has no induced $4$-cycles, then $J$ is linearly presented. For unmixed graphs without $3$- and $5$-cycles, we classify combinatorially when $J$ is linearly presented.
For projective Reed--Muller-type codes we give a global duality criterion in terms of the v-number and the Hilbert function of a vanishing ideal. As an application, we provide a global duality theorem for projective Reed--Muller-type codes over Gorenstein vanishing ideals, generalizing the known case where the vanishing ideal is a complete intersection. We classify self dual Reed-Muller-type codes over Gorenstein ideals using the regularity and a parity check matrix. For projective evaluation codes, we give a duality theorem inspired by that of affine evaluation codes. We show how to compute the regularity index of the $r$-th generalized Hamming weight function in terms of the standard indicator functions of the set of evaluation points.
In this paper we study the normality of monomial ideals using linear programming and graph theory. We give normality criteria for monomial ideals, for ideals generated by monomials of degree two, and for edge ideals of graphs and clutters and their ideals of covers.