6 problems

Let $G$ and $H$ be graphs on $n$ and $m$ vertices, respectively, with $n,m\geq 3$. Let $J_{G,H}$ be the associated binomial edge ideal, let $\ell(J_{G,H})$ denote its analytic spre…

Let $C_n$ and $C_m$ be cycles with $n,m\geq 3$, let $J_{C_n,C_m}$ be their associated binomial edge ideal, and let $\ell(J_{C_n,C_m})$ denote its analytic spread. Assume that the c…

Let $T_1$ and $T_2$ be trees on $n$ and $m$ vertices, respectively, and fix $i\in V(T_1)$ and $j\in V(T_2)$. For a positive integer $l$, define $G_1$ as the tree on $n+m+l$ vertice…

Theorem gives an inequality relating the analytic spread of a monomial ideal to the dimensions determined by its Newton polytope and supporting hyperplanes. Equality conjecture. Th…

Symbolic analytic spread conjecture. One has

Let $\mathbb{K}$ be a field, let $S=\mathbb{K}[x_1,\ldots,x_n]$, and let $I\subset S$ be an integrally closed monomial ideal. Let $\ell(I)$ denote the analytic spread of $I$. Fakha…