77 problems

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 ${\cal E}_{n,k}^0$ denote the homogeneous degree-$k$ component of the quadratic algebra ${\cal E}_n^0$, and let $n$ and $k$ be positive integers. Dimension formula conjecture.…

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…

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 $m\geq 0$,

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 $2\le i\le s-1$.

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…

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

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 $K[X^1,\dots,X^k]$ 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…

Let $k[x_1,\dots,x_n]$ be a polynomial ring, and let $I=(f_1,\dots,f_r)$ be a generic ideal whose generators satisfy $\deg f_i=d_i$. For a power series $F(T)$, write $[F(T)]$ for i…

Let $S=S(a,b,c,d,e;n)$ be the summand obtained from the WZ-seed extension of the displayed $\zeta(5)$ series, and let $V_N(S)$ be the span of the coefficients of degree-$N$ paramet…

Let $S=S(a,b,c;n)$ be the summand obtained from the WZ pair in the harmonic extension of the displayed Ramanujan $1/\pi$-series, and let $V_N(S)$ be the span of the coefficients of…

Let $S_1,S_2,S_3$ be the three summands obtained from the WZ seeds used to construct harmonic extensions of the displayed Ramanujan $1/\pi$-formula, and let $V_N(S_i)$ denote the s…

Let $S_1=S_1(a,b,c,d;n)$ and $S_2=S_2(a,b,c,d;n)$ be the two summands obtained from the WZ seeds in the harmonic extension of the displayed series, and let $V_N(S_i)$ be their degr…

Let $S_1,S_2,S_3$ be the three summands defined from the WZ identities in the preceding construction, and let $V_N(S_i)$ denote the span of the coefficients of degree-$N$ parameter…

S1 dimension conjecture. The following two assertions are conjectured:

Image Hilbert–Poincaré conjecture. The generating function of the dimensions of the images is rational, with initial expansion

Hilbert–Poincaré rationality conjecture. The Hilbert–Poincaré series of these coefficient spaces is rational:

Order-one differential-algebraicity conjecture. The Hilbert series of $\mathcal{P}$ is differential algebraic of order $1$ over $\mathbb{Z}[t]$.

Khoroshkin–Piontkovski conjecture. The Hilbert series of $\mathcal{P}$ is differential algebraic over $\mathbb{Z}[t]$.

Polishchuk–Positselski conjecture. The Hilbert series of $A$ is a rational function.

Shellability and Hilbert-series conjecture. The abstract simplicial complex corresponding to $\tilde{I^{m,n}_{r,k}}$ is shellable, and the Hilbert series of the principal component…