15 problems

Least-denominator conjecture. $\bar{C}(n,k)$ is a rational function with least denominator equal to the denominator of the $(n-1)(k-1)$-st derivative of $F(t)$.

Conjectural Poincaré–Betti series formula. The polynomial $q_n(t)$ satisfies $q_n(-1)\neq 0$, and

Diagonal-gcd conjecture. For all $n$,

Denominator-relation conjecture. For all $n$,

Vanishing-numerator conjecture. For $n\ge5$,

Binomial-denominator conjecture. The denominators $D(C_{n,2};s,t)$ and $D(T_{n,2};s,t)$ can be written as products of binomials $1-s^a t^b$, where $a$ and $b$ are nonnegative integ…

Let $n\ge3$, and write the Poincaré series in lowest terms as … where $A_n,B_n\in\mathbf Z[t]$ are relatively prime and $A_n(0)=B_n(0)=1$. Irreducibility conjecture. The polynomial…

Let $\Gamma_n=\operatorname{Sp}_n(\mathbf Z)$, let $T$ be a positive-definite index for the Siegel Poincaré series $P_{T,\Gamma_n}$, and assume the usual parity condition for scala…

Let $C_m$ be the Kunz cone, let $F$ be a face of $C_m$, and let $S$ and $T$ be numerical semigroups whose corresponding points lie in $F$. For a field $\Bbbk$, write $\Bbbk[S]$ and…

Let $G(t)$ be the rational function … whose extra numerator factor is $(1+t+\cdots+t^{n-1})^2$. Equivalently, if $G(t)=\prod_i(1-t^i)^{-\beta(i)}$, then … Here $\bar{R}(n,k)$ denot…

Zariski's multivariable Poincaré-series conjecture. The series $M_{X,D,D_1,D_2,\ldots,D_l}(t_1,t_2,\ldots,t_l)$ is rational: there exist $f,g\in{\mathbb Z}[t_1,t_2,\ldots,t_l]$ wit…

Let $(R,\mathfrak{m},k)$ be an Artinian local ring, with all modules finite. Let $[C_n]\vartriangleleft\cdots\vartriangleleft[C_1]\vartriangleleft[C_0]$ be a chain in…

Let $K^{red}(\mathcal U)$ be the Grothendieck ring of finite direct sums of indecomposable reduced injective unstable modules, and let $v$ be a virtual module in…

Let $L_\lambda$ be the indecomposable injective unstable module indexed by a partition $\lambda$, and let $P_\lambda(q)$ denote its Poincaré series. Linear independence conjecture.…

Let $R$ denote the radius of convergence of the Poincaré series of the space of long knots modulo immersions. Positive-radius conjecture. The radius of convergence is positive; tha…