121 problems

Let $S=\mathsf k[x_1,\ldots,x_n]$ be a polynomial ring, and let $R=S/I$ be a general quotient of type $(N;d_1,\ldots,d_N)$: the ideal $I$ is minimally generated by forms…

Let $A$ be a finitely presented graded algebra, and let $H_A(z)$ denote its Hilbert series. Govorov's conjecture. The Hilbert series $H_A(z)$ is rational. This conjecture is a hist…

Switching-rook-polynomial conjecture. The $h$-polynomial of $\mathbb{K}[\mathcal{P}]$ is equal to $\widetilde{r}_{\mathcal{P}}(t)$, and its regularity is equal to $r(\mathcal{P})$.

Let $G$ be a graph on $n$ vertices, let $J_G$ be its binomial edge ideal in a polynomial ring $S$, and let $h_{S/J_G}(t)$ be the numerator polynomial of the Hilbert series of…

Almost-log-concavity conjecture. The Hilbert series of $\mathbb{C}[\mathbf{x}_{n\times n}]/I_{\mathfrak{S}_{n,r}}$ is almost log-concave: log-concavity may fail for some small indi…

Anick's conjecture. Suppose $A$ is an augmented connected graded $K$-algebra with polynomial growth and with global dimension $d<\infty$. Then the Hilbert series of $A$ is given by…

Lewis–Reiner–Stanton's general-linear Hilbert-series conjecture. The Hilbert series of ${\mathcal{Q}}_m(n)^{G_n}$ is

Generic quadratic Hilbert-series conjecture. For the indicated values of $n,r$, the following formulas hold:

Let $S$ be a polynomial ring over $\mathbb C$, and let $g_1,\ldots,g_r$ be generic forms of degree $d>1$. Nicklasson's conjecture. The ideal … has the same Hilbert series as an ide…

Let ${\cal E}_n^0$ be the quadratic algebra, let $Z_{n,k}$ be the sets of normal monomials defined in the paper, and write $H(A;t)$ for the Hilbert polynomial of a graded object…

Let ${\cal E}_n^t$ be the quadratic algebra with central parameters $t_{ij}=[ij]^2$, and let $\mathbb{Z}[t]=\mathbb{Z}[t_{ij},\ 1\leq i<j\leq n]$. Finite-generation conjecture. The…

Hilbert-series conjecture. Theorems A and B are true for all three-dimensional Artin–Schelter regular algebras generated in degree one, not only for the generic ones.

Let ${\rm DQ}_n={\mathbb Q}[{\bf x},{\bf y}]/\mathcal J_n$ be the quotient by the ideal generated by diagonally quasi-symmetric polynomials without constant term, and let $M_n$ be…

Let $R=k[x_1,\ldots,x_n]$, let ${\mathfrak c}\subset R$ be a general complete intersection of type $(d_1,\ldots,d_c)$, and let $A=R/I$ be relatively compressed with respect to…

The filtered Hilbert-series conjecture. For $m\geq 0$,

Postnikov–Shapiro Hilbert-series conjecture. For every linear degree function $\phi$, the Hilbert series satisfy

Stanley's conjecture. The sequence $(h_0,h_1,\dots,h_s)$ is log-concave, meaning that $h_{i-1}h_{i+1}\le h_i^2$ for $2\le i\le s-1$.

Fomin–Kirillov's conjecture. If $\mathcal E_6$ is finite-dimensional, then its top form is unique and its Hilbert series has a symmetric increasing and decreasing form.

Let $V_n$ be an $n$-dimensional vector space, let $I_n=(f_1,\dots,f_r)$ be a generic ideal in the exterior algebra on $V_n$, and suppose that $f_i$ has degree $d_i$. Let $q_n(t)$ b…

Let $V_n$ be an $n$-dimensional vector space, let $\mathfrak{S}(V_n)$ denote the square-free algebra, and let $I_n$ be a generic ideal with generators of degrees $d_1,\dots,d_r$. W…

Let $V_n$ be an $n$-dimensional vector space, let $f$ be a generic cubic form in the exterior algebra, and let $p_{n,3}(t)$ denote the Hilbert series of the quotient by $(f)$. For…

Odd-degree annihilator correction conjecture. For odd $d>3$,

Let $K$ be a field, let $V_n$ be an $n$-dimensional vector space, and let $I$ be a generic homogeneous ideal in the exterior algebra on $n$ generators, with numerical character…

Let $K[X^1,\dots,X^k]$ be the polynomial ring in $k$ sets of countably many variables, with $\operatorname{Sym}(\infty)$ permuting the indices. Let $I\subset K[X^1,\dots,X^k]$ be a…

Let $K[X]$ be the polynomial ring in countably many variables, with the symmetric group acting by permuting variables. A homogeneous ideal $I\subset K[X]$ is symmetric if it is inv…