9 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…

The recalled conjecture. The Hilbert depth satisfies

Product bounds conjecture. The following inequalities hold:

Let $S$, $I_n$, and $J_n$ be as above, and for each $N$ let $\#\{n\leq N:\,P(n)\}$ denote the number of positive integers $n\leq N$ satisfying $P(n)$. Hilbert-depth frequency conje…

Minimality conjecture. If $q<q(d)$ or $n<n(d)$, then

Let $S=K[x_1,\ldots,x_n]$ and let $I\subset S$ be a squarefree monomial ideal. The nine-variable Hilbert-depth conjecture. If … then … The paper gives examples showing failure of t…

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 $S=\mathbb{K}[x_1,\dots,x_n]$, and let $I\subset S$ be a lexsegment ideal generated by monomials of degree $d$. Write $\mu(I)$ for the number of minimal generators of $I$. Lexs…

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…