5 problems

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 $I\subset S$ 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…

Let $S=K[x_1,\dots,x_n]$ be a polynomial ring over a field $K$. A monomial ideal $I\subset S$ is componentwise polymatroidal when each component $I_{\langle j\rangle}$ is polymatro…

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…