100 problems

Fibonacci degree conjecture. Both polynomial expressions $\gamma_n(b_1^{m_1}\dots b_k^{m_k})$ and $\ell_n(b_1^{m_1}\dots b_k^{m_k})$ have degree $F_n$ in $m_j$ and degree $F_{n-1}$…

Distinct-root conjecture. One has $\rho_1\neq\rho_2$. This asserts that distinct admissible parameter triples determine distinct spectral radii for the corresponding Type $5$ digra…

Let $\ell=2$, i.e., $x=(x_1,x_2)$, and let $d\in\mathbb N_0$. Suppose $p\in\mathbb R[x,y]$ and $q\in\mathbb R[x,z]$ are real zero polynomials of degree at most $d$ such that…

Let $f(z)=p(z)\overline{q(z)}$ be a logharmonic polynomial, where $p$ and $q$ are analytic polynomials of degrees $n$ and $m$, respectively, with $p$ not a constant multiple of $q$…

Let $R(x)=1+\sum_{i\geq 2}R_ix^i$, and define $C_m$ by $C_0=1$, $C_1=0$, and … For $n\geq 0$, let $\Sigma_{k,2n}$ be the sum of the terms of weight $k+1-2n$ in $\Sigma_k$. A polyno…

For $n\geq 0$, let $\Sigma_{k,2n}$ denote the component of weight $k+1-2n$ in the Kerov character polynomial $\Sigma_k$, and write $[R_2^i]\Sigma_{k,2n}$ for the coefficient of…

For integers $d$ and $0<k<m<d$, let $P_{k;d}(x)$ and $P_{m;d}(x)$ denote the truncated binomial polynomials used in the paper. Common-root conjecture. The polynomials $P_{k;d}(x)$…

Simplicity conjecture. All nonzero roots of $W_N^{[m,k,l]}(z)$ are simple.

Let $\mathcal B(n)$ be the set of binary partitions of $n$, and let $\mathrm{num}_{\mathcal B}(n,x)$ be the associated numerator polynomial. Binary-partition nondivisibility conjec…

Let $\mathcal B(n)$ be the set of binary partitions of $n$, and let $\mathrm{num}_{\mathcal B}(n,x)$ be the associated numerator polynomial. Binary-partition coefficient conjecture…

Let $\mathcal B(n)$ be the set of binary partitions of $n$, and let $\mathrm{num}_{\mathcal B}(n,x)$ and $\mathrm{den}_{\mathcal B}(n,x)$ denote the associated numerator and denomi…

Let $\mathrm{den}(n,x)$ be the denominator polynomial associated with sums of reciprocals of subsum polynomials over partitions of $n$. Log-concavity conjecture. The sequence of co…

Let $\mathrm{num}(n,x)$ be the numerator polynomial associated with sums of reciprocals of subsum polynomials over partitions of $n$, and write … where $\mathrm{num}_0(n,x)$ is eve…

Let $n\geq 1$, and let $\mathrm{num}(n,x)$ and $\mathrm{den}(n,x)$ be the reduced numerator and denominator associated with the sum of reciprocals of subsum polynomials over partit…

Let $n\geq 1$, and let $\mathrm{num}(n,x)$ denote the numerator polynomial associated with sums of reciprocals of subsum polynomials over partitions of $n$. Irreducibility conjectu…

Visibility Density Conjecture. If $F(x)\in\mathbb{Z}[x]$ has at least two distinct roots, then the set of lattice points in $\mathbb{Z}_{>0}\times\mathbb{Z}_{>0}$ visible along…

Erdős–Turán conjecture for two-variable polynomials. There exists a constant $c$ depending only on $f$ such that

Subgroup-counting positivity conjecture. If $\ell=3$ or $\ell=4$, then for every $n\geq a_{\ell}$ and every real $x\geq 1$,

D'Arcais positivity conjecture. For every integer $n\geq 6$ and every real $x\geq 1$,

Reducibility conjecture. If $x^{\underline{k}}-a!$ is reducible over $\mathbb{Z}$, then $(k,a)=(3,6)$, $(k,a)=(4,7)$, or there exists an integer $t$ such that $a=t!-1$ and $k=t!-t$…

Let $P_n(z)$ be the sorted binomial polynomial. For odd $n=2m+1$, the source identifies the relevant target group as the $m$th hyperoctahedral group. Hyperoctahedral Galois group c…

Let $P_n(z)$ be the sorted binomial polynomial with parameter $n$. Irreducibility conjecture. $P_n(z)$ is irreducible over $\mathbb{Q}$ for all even $n$. The analogous factorizatio…

Let $P_n(z)$ denote the sorted binomial polynomial, and call a root nontrivial when it is not the trivial root shared by the relevant polynomials. Common-root conjecture. $P_n(z)$…

The four matroid polynomial conjectures. For every matroid $\mathsf{M}$, the polynomial $P_{\mathsf{M}}(x)$ is real-rooted, the polynomial $Z_{\mathsf{M}}(x)$ is real-rooted, the p…

For $n\geq 1$, let $P_n^{\sigma}(x)$ be the D'Arcais polynomial defined by … A polynomial is Hurwitz if the real parts of all its zeros are negative. Heim–Neuhauser's conjecture. T…