15 problems

Let $S$ be a polynomial ring and let $I\subset S$ be a monomial ideal. For every monomial prime ideal $P\subset S$, let $I(P)$ denote the monomial localization of $I$ at $P$, obtai…

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 $R=K[x_1,\ldots,x_n]$ 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 $I=I_{\mathbf a,d}$ be the Veronese-type polymatroidal ideal, with the notation $\mathbf a$ and $d$ used in the paper. For $i=1,\ldots,n-1$, let $\operatorname{HS}_i(I)$ denote…

Let $S$ be the polynomial ring in the paper, let $I\subset S$ be a polymatroidal ideal, and let $\operatorname{HS}_i(I^k)$ denote the $i$th homological shift ideal of $I^k$. Associ…

Let $I$ be a polymatroidal ideal, and let $\mathcal{R}(I)$ denote its Rees algebra. For each $i>0$, let $\operatorname{HS}_i(\mathcal{R}(I))$ be the $i$th homological shift algebra…

Let $S=K[x_1,\dots,x_n]$ be a polynomial ring over a field $K$, and let $I\subset S$ be a polymatroidal ideal. For $k\ge 1$, write $I^{(k)}$ for the $k$th symbolic power of $I$. Po…

Let $I\subset S$ be a componentwise polymatroidal ideal in a polynomial ring $S=K[x_1,\dots,x_n]$. Power conjecture. Every power $I^k$ of $I$ has linear quotients. The source gives…

Herzog–Qureshi's conjecture. Does

Let $K$ be a field, $S=K[x_1,\dots,x_n]$, and let $I\subset S$ be a polymatroidal ideal. The socle ideal $\operatorname{soc}(I)$ is the unique monomial ideal satisfying…

Let $R=K[x_1,\ldots,x_n]$ be a polynomial ring over a field, and let $I$ be a polymatroidal ideal of $R$. Write $\astab(I)$ for the index at which the sets of associated primes…

Let $S=\mathbb{K}[x_1,\dots,x_n]$, let $I$ be a polymatroidal ideal, and let $\ell(I)$ denote its analytic spread. Asymptotic Stanley-depth conjecture. For every sufficiently large…

Reverse-lexicographical characterization conjecture. The ideal $I$ is polymatroidal if and only if it has linear quotients with respect to the reverse lexicographical ordering of i…

Let $S=K[x_1,\ldots,x_n]$. A monomial ideal $I\subset S$ is polymatroidal if its minimal monomial generators are indexed by the bases of a polymatroid. For each monomial prime idea…

Let $S=K[x_1,dots,x_n]$ and let $I\subset S$ be a polymatroidal ideal. Write $\astab(I)$ for the index of stability of the associated primes of the powers of $I$, and $\dstab(I)$ f…