16 problems

Positivity conjecture. For $d\geq 2$, the trigonometric polynomials $\mathcal T_n^{(d)}(t)$ are positive for every $t\in\mathbb R$. Positivity of these polynomials is suggested as…

Let $d>2$ and define the coefficients $b_k^{(d)}$ by … For each $n\geq 0$, the corresponding partial sum is the polynomial $\sum_{k=0}^n b_k^{(d)}z^k$. Zero-free Taylor polynomial…

Let $I$ index vectors $\boldsymbol{v}_{ij}$, let $K_0$ be the monoid used to define the formal power series $\boldsymbol{z}^{\boldsymbol{k}}$, and let $\boldsymbol{v}_{ij}$ be Fano…

Non-negative coefficient conjecture. For each integer $r\geq 1$, this power series has all non-negative coefficients.

Clark–Ismail's coefficient-positivity conjecture. Then $\gamma(j,k)\ge0$ for all $j>1$ and $k\ge0$. This is a coefficientwise strengthening of the positivity question for $g_j$ and…

Radius-of-convergence conjecture. There exist integers $l_1,\ldots,l_m$ such that the radius of convergence of this series is

Let $p$ be a prime and let $r\ge0$. Define … and let … be the associated Böttcher coordinate. Asymptotic valuation and radius conjecture. For $k\equiv0\pmod p$, … Moreover, the rad…

Let $t_m(n)$ denote the coefficient sequence associated with the power series in the paper, and let $m\in\mathbb{N}_{\geq 2}$. Unboundedness conjecture. … The preceding estimate gi…

Polynomial ramification criterion. For every positive integer $b$, there exists a polynomial $P_b\in\mathbb{Z}[x_1,\dots,x_{b+1}]$ such that $f$ is $b$-ramified if and only if…

Rationality–transcendence conjecture. The power series $F_V(x_1,\ldots,x_d)$ is either rational or transcendentally transcendental.

The radius-of-convergence conjecture. The radius of convergence of this series is $\pi^2$. The preceding results establish convergence for a positive range of $x$ and divergence fo…

Let $N_1,N_2,\ldots,N_k\ge2$ be integers, let $n$ be a positive integer, and let ${\bf q}_{\bf N}(z)$ be the function associated with the mirror map. For a power series, write…

Let $N_1,N_2,\ldots,N_k$ be positive integers, all at least $2$, and let ${\bf z}_{\bf N}(q)$ be the corresponding mirror map. Sign-pattern conjecture. 1. If $\Phi_{\mathbf N}\le3$…

Let $N_1,N_2,\ldots,N_k\ge2$ be integers with $\Phi_{\mathbf{N}}\ge4$, and let ${\bf z}_{\bf N}(q)$ be the mirror map with associated function ${\bf q}_{\bf N}(z)$ and constant…

2- and 3-adic expansion conjecture. The polynomials $\gamma_{-\frac{1}{2}+r}^{\ast\ast}(a)$ possess 2- and 3-adic expansions to all orders:

Zudilin's conjecture. For any positive integers $N_1,N_2,\dots,N_k$,