41 problems

Finite-part factorization conjecture. There are positive integers $d_1,d_2,\ldots,d_l$, with $l\leq n$, such that

Reciprocal-polynomial realization conjecture. Any reciprocal polynomial having negative roots and leading coefficient equal to $1$ should have the form $P_\Lambda(t)$ for some even…

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

Stabilized-difference conjecture. Recall the integers $c_{n,2}^\bullet(k)$ introduced in Appendix B. We conjecture that, for each fixed $k\ge0$,

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 $n\ge3$, write $n=2s-1$ when $n$ is odd and $n=2s$ when $n$ is even, and let $r_n(t)$ be the polynomial defined in the paper. Write the Poincaré series in lowest terms as … whe…

Let $A=k[x_1,\ldots,x_n]$ be a polynomial ring over a field $k$, let $I$ be a homogeneous ideal, and set $R=A/I$. The Poincaré series of $R$ is … where…

Let $^5\mathcal{P}$ be the operad discussed in the paper, and let its Poincaré series be the formal power series whose coefficients encode the dimensions of its components. Conside…

Let $S=k[x_1,\ldots,x_n]$ and let $\mathfrak a=\langle m_1,\ldots,m_l\rangle$ be a monomial ideal with minimal monomial generating system. Write $A=S/\mathfrak a$,…

Let $\mathcal{A}_{n,r}$ be a generic $n$-arrangement with $r$ hyperplanes, where $3\leq n$ and $r\geq m+n$. Let $\Xi^{(m)}(\mathcal{A}_{n,r})$ be the submodule appearing in the dec…

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 $P_{k,m}$ denote the Poincaré series of level one and weight $k$, where $m$ is a positive integer, and suppose that the corresponding space of cuspforms is nontrivial. Poincaré…

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…

Let $X$ be a smooth special rigid $K$-variety, let $\omega$ be a gauge form on $X$, and let $P(X,\omega;T)$ denote its volume Poincaré series. Write $d$ for the dimension of $X$, a…

Serre–Kaplansky conjecture. The Poincaré series $P_A(t)$ is a rational function.

For a generalized Cartan matrix $A$ with index set $I$ of cardinality $n$, let $K(A)$ be the associated Kac-Moody group and let $F(A)$ be its flag manifold. Write its Poincaré seri…

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…

Geometric factorization conjecture. There exist $T_1,\ldots,T_{n-1}\in W$ such that each $T_i$ is a geodesic-tube generator of length $d_i$, and, on writing

Let $\mathcal{P}_W(T)$ and $\mathcal{P}_{\rm exc}(T)$ be the two Poincaré series associated with an $n$-fold cover of a connected reductive group $G$. In the examples discussed, bo…

Infinite-factor conjecture. There are positive integers $j_1,j_2,\ldots,j_k,\ldots$ such that