226 problems

Conforti–Cornuéjols conjecture. All symbolic powers and ordinary powers of $I$ coincide if and only if $I$ has the packing property:

The recalled conjecture. The Hilbert depth satisfies

Let $\Gamma=(\mathsf{V},\mathsf{E})$ be a connected graph, let $\overline{\mathsf{E}}$ denote the relevant complementary edge set, and let $n$ be the number of vertices. Define … w…

Küronya–Pintye conjecture.

Let $I=\langle f_0,f_1,\ldots,f_{p-1}\rangle\cap R^{S_n}$ be the elimination ideal associated with the transfer ideal, and let $A$ be the block matrix whose maximal minors generate…

For a monomial ideal $I$ in a polynomial ring $S$, write $\HS_i(I)$ for its $i$-th homological shift ideal, defined by … An ideal is polymatroidal when its minimal monomial generat…

Let $G$ be an unmixed graph, meaning that all minimal vertex covers of $G$ have the same cardinality, and let $J(G)$ be its vertex cover ideal. Recall that a graph is vertex decomp…

Let $S$ be a polynomial ring and let $I\subseteq S$ be a monomial ideal with linear powers, meaning that $I^k$ has a linear resolution for every $k\geq 1$. Let $\alpha(I)$ denote t…

Let $P$ and $Q$ be posets with $s$ and $t$ connected components, respectively. Write $\mathfrak{a}(P)$ and $\mathfrak{a}(Q)$ for the invariants appearing in the proposed formulas,…

Let $S$ be a polynomial ring and let $I$ be a monomial ideal in $S$. An ideal is Simis when its symbolic Rees algebra is generated by its elements of degree one; a monomial ideal h…

Fix integers $d,m$ with $d\geq m\geq 1$, and suppose that $\alpha$ satisfies … Let $\Delta\sim\Delta_m(n,n^{-\alpha})$ be the corresponding multiparameter random simplicial complex…

Let $c=1$, so the polynomial ring has one row of variables, and let ${\mathcal I}=(I_n)_{n\ge 1}$ be a nonzero $\operatorname{Inc}$-invariant chain of proper monomial ideals. Here…

Let $S=\mathbb{K}[x_1,\dots,x_n]$ and let $\mathfrak{m}=(x_1,\ldots,x_n)$ be its maximal ideal. For a monomial ideal or module, write ${\rm sdepth}$ for Stanley depth. Cimpoeaş's c…

Vasconcelos's conjecture. The Rees algebra $\mathcal{R}(I)$ is almost Cohen--Macaulay, meaning

Let $R=K[x_1,\ldots,x_n]$ be a polynomial ring over a field, let $I\subset R$ be a polymatroidal ideal, and write $\astab(I)$ and $\dstab(I)$ for its indices of associated-prime an…

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 $G=(V,E)$ be a simple graph, where $V$ is its vertex set. A graph is critically $s$-chromatic if it has chromatic number $s$ and deleting any vertex lowers its chromatic number…

Depth criterion conjecture. One has

Let $K$ be a field, let $S=K[x_1,\dots,x_n]$, and let $I_{n,1}=(x_1,\dots,x_n)$ be the squarefree Veronese ideal of degree $1$. Its Stanley depth is denoted by…

Let $k$ be a finite field with $|k|=q$, let $X$ be a set of state transition pairs, and let $M_X$ be the associated monomial ideal with generating set $W$. A monomial tree is the d…

Let $S$ be a polynomial ring, let $I_{\mathcal L}$ be a lattice ideal in $S$, and let $M$ be its initial ideal with respect to some term order. For a prime ideal $P$ in $S$, write…

Algebraic Conforti–Cornuéjols conjecture. If

Let $R$ be the polynomial ring in $x$ and $y$, and for any $d$ let … An ideal is Ratliff–Rush if it equals its Ratliff–Rush closure. HJLS conjecture. For any $d$, the ideal $I_d$ a…

Let $T$ be a forest, let $I(T)$ be its edge ideal, and let a tree-like system for $I(T)$ be a system whose support is the set of edge monomials of $I(T)$. For stretched forests, th…

Let $R=K[x_1,\ldots,x_n]$, and let $I$ be a squarefree monomial ideal. Write $\operatorname{ara} I$ for its arithmetical rank and let $\tau(I)$ be the maximum height of a minimal p…