14 problems

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 $K$ be a field, let $S=K[x_1,\ldots,x_n]$, and let $I\subset S$ be a squarefree monomial ideal. For a finitely generated graded $S$-module $M$, write $\operatorname{hdepth}(M)$…

Uniform boundedness conjecture. For every $n\geq 1$, there exists a constant $C(n)>0$ such that

Approximate Hilbert-depth conjecture. The Hilbert depth is expected to satisfy

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 $S=K[x_1,\ldots,x_n]$ and $S'=K[x_1,\ldots,x_m]$ be polynomial rings. Let $I$ be a strongly stable monomial ideal in $S$ generated by monomials of the same degree, and let…

Let $S$ be a polynomial ring, and let $L_1$ and $L_2$ be lex ideals in $S$, both generated by monomials of degree $d$, with $L_1\subset L_2$. Write $\operatorname{hdepth}(L)$ for t…

Let $S=K[x_1,\ldots,x_n]$ be a polynomial ring, and let $I$ be a stable ideal generated by monomials of degree $d$. Write its Hilbert series as … Here $\operatorname{hdepth}(I)$ de…

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…

Let $S=\mathbb{K}[x_1,\dots,x_n]$ and let $I\subset S$ be a stable ideal generated in degree $d$. Write $H(I,k)$ for the Hilbert function of $I$ in degree $k$, and let…