131 problems

Let $S$ be the polynomial ring and $I_{n,n}$ the special class of squarefree monomial ideals considered in the paper, and let $\operatorname{hdepth}(S/I_{n,n})$ denote the Hilbert…

Let $\Delta_{m,n}$ be a chessboard complex with $n\geq m\geq 1$, and let $S/{\mathcal F}(\Delta_{m,n})$ denote its facet ring. Regularity-depth equality conjecture. … The preceding…

Let $R=K[x,y,z]$, and let … be a level Artinian almost complete intersection, where $t>0$ and $0\leq\alpha\leq\beta\leq\gamma$. Assume that $K$ is algebraically closed of character…

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…

Let $S$ be a polynomial ring, let $\succ$ be a term order on $S$ other than the graded reverse lexicographic order, and let $k\geq 2$ be the smallest degree on which $\succ$ and th…

Let $k[\mathbf{x}]=k[x_1,\ldots,x_n]$, and let $F$ be a finite set of $q\geq n$ monomials of the same degree $d\geq 2$. Suppose that the ideal $(F)\subset k[\mathbf{x}]$ is of resi…

Let $S=k[x_1,\ldots,x_n]$, let $\mathfrak a=\langle m_1,\ldots,m_l\rangle\subset S$ be a monomial ideal, and set $A=S/\mathfrak a$. The ideal $\mathfrak a$ satisfies the strong gcd…

Let $S=k[x_1,\ldots,x_n]$, let $\mathfrak a=\langle m_1,\ldots,m_l\rangle$ be a monomial ideal, and set $A=S/\mathfrak a$. Let $R$ be the graded commutative algebra constructed fro…

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…

The recalled conjecture. The Hilbert depth satisfies

Let $I\subset k[x_1,\dots,x_n]$ be a square-free monomial ideal generated in degree $d$ and having a linear presentation. Let $G_I$ be the graph whose vertices are the minimal mono…

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.

Phase transition conjecture. The $\lcm$-density of edge ideals of graphs exhibits a phase transition phenomenon when considering the set of all graphs on $n$ vertices as…

The v-number bound. For every simple connected graph $G$,

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

Let $I\subset K[x_1,\ldots,x_n]$ be a monomial ideal, and let $h$ be its big height. Let $\alpha(I)$ be the least degree of a nonzero element of $I$, and let … be its Waldschmidt c…

Bonanzinga–Eliahou's conjecture. For $n\geq 3$, the Gotzmann threshold $\tau_n(x_2^d)$ is a polynomial of degree $2^{n-2}$ in $d$ with the same dominant term as the $(n-2)$-iterate…

Let $\mathbb K$ be a field, let $R=\mathbb K[x_1,\ldots,x_n]$, and let $I\subseteq R$ be a Cohen--Macaulay monomial ideal. Monomial glicci conjecture. The ideal $I$ is glicci, mean…

Stirling-number Lech conjecture. The first displayed inequality, together with the equivalent reindexing given in the source, holds for every $\mathfrak{m}$-primary monomial ideal…

Castillo–Cid-Ruiz–Mohammadi–Montaño conjecture. The Möbius support of $\mathscr{P}$ is a generalized polymatroid, that is, a homogenization of it yields a polymatroid. The paper se…

Let $R=\mathbb{k}[x_1,\ldots,x_p]$ be a polynomial ring over a field $\mathbb{k}$, and let $\mathscr{P}$ be a polymatroid on $[p]$ with cage $\mathbf{m}\in\mathbb{N}^p$. Let…

Dominance threshold conjecture. As $d\to\infty$, for every fixed $p>0.1$, the probability that the random homogeneous ideal is dominant tends to $0$.

For an integer $q\geq 1$, let $\mathcal{S}_q=\mathsf{k}[x_A\colon \emptyset\neq A\subseteq [q]]$, and let $\mathcal{E}_q$ be the ideal generated by … For $r\geq 1$, write…

Let $I$ be a square-free monomial ideal. Eventual stability conjecture. Then, $\operatorname{assrad} I^s$ is constant for $s\gg 0$. This is the ordinary-power analogue of the event…