19 problems

Let $S=K[x_1,\ldots,x_n]$, let $J_{n,m}$ be the path ideal of the cycle graph on $n$ vertices with paths of length $m$, let $d$ be the parameter defined by the paper for $n$ and…

Let $K$ be a field, let $S=K[x_1,x_2,\ldots,x_n]$, let $I\subset S$ be a proper monomial ideal with linear quotients, and write $[n]=\\{1,\ldots,n\\}$. The stronger variable reduct…

Let $G$ be a graph with $n$ vertices, let $J(G)$ be its cover ideal, and let $J(G)^{(k)}$ denote its $k$-th symbolic power. Let $\nu_o(G)$ denote the parameter used in the source f…

Let $S=\mathbb{K}[x_1,\dots,x_n]$ and let $I$ be a monomial ideal of $S$. Denote the Stanley depth of a module by ${\rm sdepth}$. Herzog's convergence conjecture. The sequence … is…

Let $S$ be a polynomial ring over a field and let $I\subset S$ be a monomial ideal. Write $\operatorname{depth}$ for homological depth and $\operatorname{sdepth}$ for Stanley depth…

Let $R=\mathds{K}[X_1,\ldots,X_n]$ be a polynomial ring in an arbitrary positive number of variables, let $\mathfrak m=(X_1,\ldots,X_n)$ be its homogeneous maximal ideal, and let…

Let $S=K[x_1,\ldots,x_n]$ be a polynomial ring, and let $J_n$ denote the ideal associated with the cyclic graph on $n$ vertices as in the paper. The Stanley depth conjecture. For a…

Lattice gluing conjecture. If

Lattice Stanley projective dimension conjecture. For all finite lattices $L$, the following inequalities hold:

Lattice formulation of Stanley's conjecture. For all finite semilattices $L$, the following inequalities hold:

Let $S=K[x_1,\ldots,x_n]$. Let $I\subset S$ be minimally generated by squarefree monomials $f_1,\ldots,f_r$ of degree $d$, and let $E$ be a set of squarefree monomials of degree at…

Let $R=\mathbb{K}[X_1,\ldots,X_n]$ be a polynomial ring, let $I\subset R$ be a monomial ideal, and let $I^p\subset R^p$ be its polarization, where $R^p$ is the polynomial ring in w…

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…

Let $S=\mathbb{K}[x_1,\dots,x_n]$, and let $I\subset S$ be a squarefree strongly stable ideal generated by monomials of degree $d$. Let $I^\tau$ be the monomial ideal obtained from…

Let $S=\mathbb{K}[x_1,\dots,x_n]$ and let $I\subset S$ be an integrally closed monomial ideal. If $\ell(I)$ denotes the analytic spread of $I$, then Seyed Fakhari's conjecture. … T…

Let $R=K[x_1,\cdots, x_n]$ be a polynomial ring in $n$ variables. Let $I_{n,d}$ be the squarefree Veronese ideal generated by all squarefree monomials of degree $d$, and let…

Let $K$ be a field, let $S=K[x_1,\dots,x_n]$, and let $I_{n,d}$ be the ideal generated by all squarefree monomials of degree $d$ in $S$. For positive integers $d$ and $n$ with…

Let $K$ be a field, $S=K[x_1,\ldots,x_n]$, and let $J\subset I$ be monomial ideals. Let $P_{I/J}$ be the characteristic poset of $I/J$, and let $\operatorname{reg}(I/J)$ denote the…

Let $S=K[x_1,\ldots,x_n]$ and let $M$ be a finitely generated ${\mathbb Z}^n$-graded $S$-module. A Stanley decomposition of $M$ is a finite direct sum … Here $m_iK[Z_i]$ denotes a…