29 problems

Guo–Yang polynomiality conjecture. For each $k$, the coefficient of $g^{-k}$ in this expansion is a polynomial in $n$ and in the multiplicities of the arguments, and only the multi…

Anand–Dumir–Gupta conjecture. The function $H_n(r)$ is a polynomial in $r$ of degree $(n-1)^2$ with certain additional properties.

For each positive integer $m$, let ${\mathsf C}$ be the operator and let $\Lambda$, $\xi_m^{(i)}$, and $F_m^g$ denote the series and polynomials defined in the source. Polynomialit…

Let $\mathfrak{o}$ be a discrete valuation ring with maximal ideal $\mathfrak{p}$ and residue-field cardinality $q=|\mathfrak{o}/\mathfrak{p}|$. For a partition $\lambda\in\Lambda$…

Polynomiality conjecture. (a) The functions ${\mathcal P}_n(x,z)$ are polynomials in $x,z$ of degree $n$ in $x$; each $r_i^{(n)}(z)$ is a polynomial in $z$ satisfying

Let $\mathfrak{g}^F$ be a finite Lie algebra, and call two elements $X,Y\in\mathfrak{g}^F$ of the same type when their centralisers $C_G(X)$ and $C_G(Y)$ are conjugate. Polynomiali…

Strong conjecture. The class $c_k\big(\Pol^d(\mathbb{C}^n)\big)$ is of the form $P(B_1,\dots,B_s)$, where $P$ is a polynomial. This strengthens the weak conjecture by proposing an…

Let $\Pol^d(\mathbb{C}^n)$ denote the relevant polynomial-space bundle, and let $c_k\big(\Pol^d(\mathbb{C}^n)\big)$ be its $k$th Chern class. Weak conjecture. For any fixed $k$ and…

Let $\Pol^d(\mathbb{C}^n)$ denote the relevant polynomial-space bundle, and let $c_k\big(\Pol^d(\mathbb{C}^n)\big)$ be its $k$th Chern class. Separate polynomiality conjecture. For…

Type-$A_n$ polynomiality conjecture. The cardinality of the lower Bruhat interval is a polynomial in the coordinates $\lambda_1,\dots,\lambda_n$ of degree $n$:

Let $P_g$ denote the genus-$g$ potential, and let $v_1,v_2,v_3$ be the particular elements of the ring $\mathscr{R}$ considered by Yamaguchi and Yau. Yamaguchi–Yau polynomiality co…

Let $S$, $\beta$, $\delta_i$, and the normalized descendent invariant be as in the polynomiality conjecture, with $k_i,m_i\geq1$. For subsets $I_x\subset\\{1,\ldots,x\\}$, where…

Let $S$ be a K3 surface, let $\beta\in H_2(S,\mathbb Z)$ be primitive and effective with $\beta^2\ne0$, and let $\delta_1,\ldots,\delta_t\in H^2(S)$ satisfy $\delta_i\cdot\beta=0$.…

Let $\vec{i}$ be a tuple of nonnegative integers, with $|\vec{i}|$ denoting the sum of its entries. Let $D_{(\vec{i}),2g+2}$ and $d_{(\vec{i}),2g+2}$ denote the corresponding…

Let $\operatorname{\mathfrak{q}}^C$ be a standard parabolic contraction in type $C$, and let $\operatorname{Sy}(\operatorname{\mathfrak{q}}^C)$ denote its algebra of semi-invariant…

Let $DH_{g,n}(\mu_1,\ldots,\mu_n)$ be the double Hurwitz numbers, and let $d$ and $A_\mu^i$ be defined by … where $\lambda$ is an integer partition with $\ell(\lambda)$ parts and…

Let $G(x,r)$ be the generalized generating function used in the paper, and write $H_n(G(x,r))$ for the Hankel determinant of its coefficients. For a positive integer $t$, consider…

Let $C(x)=\sum_{n\geq 0}C_nx^n$ be the Catalan generating function, let $F(x,r)=C(x)^r$, and write $H_n(F(x,r))$ for the $n\times n$ Hankel determinant of the coefficients of…

Let $\ell_i$ denote the $i$-th Lecture Hall polynomial, constructed in the paper as a Laurent polynomial depending on $y_1,\dots,y_i$. Polynomiality conjecture for Lecture Hall pol…

Let $\widetilde{\mathcal{F}}^{\mathsf{B}}_g$ be the genus-$g$ series associated to the formal quintic, let $T=I^{\mathsf{Q}}_1(q)/I^{\mathsf{Q}}_0(q)$ be the mirror coordinate, and…

Polynomiality conjecture. For the indicated integers $k$ and $j$, $S_{k,j}(n)$ is expressed as a polynomial in

Let $p$ be a prime, and let the numbers under consideration be the counts of configurations of pairs in finite abelian $p$-groups studied by Anilkumar and Prasad. Anilkumar–Prasad…

Fix $\delta\ge1$, and write the refined plane Severi degree as … Let $N_\delta(d;y)$ be the refined node polynomial, with coefficients $p^\delta_i(d)$ defined analogously. Coeffici…

Fix partitions ${\lambda}/{\mu}$ and ${\nu}$. Let $\psi_k:GT_{k{\lambda}/k{\mu},k{\nu}}\to GT_{{\lambda}/{\mu},{\nu}}$ be the map sending a pattern to the unique lexicographically…

Fix partitions ${\lambda}/{\mu}$ and ${\nu}$, let $G\in GT_{{\lambda}/{\mu},{\nu}}$, and define … Here $GT_{{\lambda}/{\mu},{\nu}}$ is the set of relevant Gelfand–Tsetlin patterns…