18 problems

Let $H(t,X)\in\mathbb Q[t,X]$ be a homogeneous polynomial of degree $n$ that is not a proper power. Consider the specialization $H(\bar t,X)-1$ for integers $\bar t$. Even-degree i…

Non-square conjecture. For every $n\ge3$, both $b_n^+(c)$ and $b_n^-(c)$ are non-square in $\mathbb{Z}[i]$.

Second-iterate irreducibility conjecture. If $f_r^2(x)$ is irreducible over $K$, then $f_r$ is stable over $K$.

Generalized Dumas–Eisenstein irreducibility conjecture. The polynomial

Harrington's conjecture. The polynomial $f$ is irreducible unless $f=x^2+4x+4$. The paper states that Zhang and Yuan provide an alternative proof using the Newton polygon technique…

Let $f=a_0+a_1x+\cdots+a_nx^n\in\mathbb Z[x]$, where $a_0=\pm p^k$ for a prime $p$ and positive integer $k$. Suppose that $a_1=a_2=\cdots=a_{q-1}=0$ for a prime $q\leq n$. Koley an…

Let $p$ and $q$ be distinct odd primes and set $n=pq$. Let $f_n$ and $g_n$ denote the reciprocal Fekete polynomial and its trace polynomial, respectively. Semiprime Fekete irreduci…

Fix an integer $d\geq 2$, let $m\geq 2$ and $n\geq 1$, and let $\zeta\neq 1$ be a $d$-th root of unity. For the Misiurewicz polynomial $G_{d,m,n}^{\zeta}(c)$ defined by the paramet…

Veselov's conjecture. This Wronskian has simple zeros, except possibly at $x=0$.

Let $l\ge3$, and let $\widetilde C_{l,0,0}(X)$, $\widetilde C_{l,0,1}(X)$, $\widetilde C_{l,1,0}(X)$ and $\widetilde C_{l,1,1}(X)$ be the four polynomial families introduced in the…

Let $C_{X,\lambda_2}$ denote the polynomial discussed in the concluding remarks, with coefficients in ${\mathbb Q}$. Irreducibility conjecture. For even $\lambda_2\ge2$,…

Let $\phi_c(x)=x^2+c$ for $c\in\mathbb{Z}$, and let $b\in\mathbb{Z}$ be any basepoint. The polynomial $\phi_c^n(x)$ denotes the $n$th iterate of $\phi_c$. Third-to-fourth iterate i…

Let $h_{\mathrm{Bin},K,d}(x)=x^d+a_{d-1}x^{d-1}+\dots+a_1x+a_0$ be a random monic polynomial whose coefficients $a_0,\dots,a_{d-1}$ are independently chosen from $[0,K]$ according…

Let $g_{\pm1,d}(x)=x^d+a_{d-1}x^{d-1}+\dots+a_0$ be a monic polynomial of positive degree $d$, where each $a_i$ is independently $+1$ or $-1$ with probability $\frac{1}{2}$. Radema…

Let $g_{\pm1,d}(x)=x^d+a_{d-1}x^{d-1}+\dots+a_0$ be a monic polynomial of odd positive degree $d$, where each $a_i$ is independently $+1$ or $-1$ with probability $\frac{1}{2}$. Ra…

Jones's polynomial Cunningham-chain reducibility conjecture. The polynomial is reducible over $\mathbb{Q}$ if and only if $j=t$. This conjecture proposes an explicit family of init…

Let $B_n(t)$ denote the Stern polynomial indexed by the positive integer $n$. For a positive integer $n$, say that it has exactly $k$ prime divisors, with the source's convention f…

Let $B_p(t)$ be the Stern polynomial indexed by a prime number $p$. The prime-index irreducibility conjecture. For every prime number $p$, the polynomial $B_p(t)$ is irreducible. T…