17 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…

Moreno-Socías's conjecture. The ideal $I$ is weakly reverse lexicographical.

Initial-ideal containment conjecture. Then

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

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 an infinite field, let $R=k[x_1,\ldots,x_n]$, and let $I$ be an ideal generated by generic forms. Let $R_j$ denote the degree-$j$ component of $R$, and let $g_{d,d'}(x)$…

For generic linear forms $l_i$, let … and let … Here $F_{n,r,d}(t)$ is the Hilbert series conjectured by Fröberg for generic degree-$d$ forms in the cases under consideration. The…

Stabilization-threshold conjecture. The remaining stated values are

Let $n,r,d$ be positive integers. Let $l_1,\ldots,l_r$ be generic linear forms and let $f_1,\ldots,f_r$ be generic forms of degree $d$ in $\mathbb{C}[x_1,\ldots,x_n]$. Iarrobino's…

Exterior-algebra expected-series conjecture. For $n-d=4,5,7,8,9$ and some larger values of $n$, the Hilbert series of $\Lambda(V)/(f)$ is the expected series; the source also recor…

Bigraded Fröberg conjecture. The Hilbert series of $R/(f_1,\ldots,f_r)$ is

Fröberg–Lundqvist conjecture. If $r\ge n$, then

Iarrobino's conjecture. Outside these exceptions, the algebras

Let $R=\mathbb{C}[x_1,\ldots,x_n]$, with $g_1,\ldots,g_k$ generic forms of degree $d\geq2$, and let $m\in\mathbb{N}$. Set … For a power series $F(t)$, write $\lceil F(t)\rceil$ for…

Let $R=\mathbb{C}[x_1,\ldots,x_n]$, and let $g_1,\ldots,g_k$ be generic forms of degree $d$. For a power series $F(t)=\sum_{i\geq 0}a_it^i$, define $\lceil F(t)\rceil$ by retaining…

Let $S=\Bbbk[x_1,\ldots,x_n]$, let $J\subset S$ be generated by generic forms, and let $f\in S$ be a generic form of degree $e$. Recursive Fröberg conjecture. For every integer $j$…