29 problems

Let $A$ be a random reciprocal polynomial in the restricted coefficient model, with coefficients chosen independently and uniformly from … and condition on $A(0)=1$. Odlyzko–Poonen…

Let $q\geq 3$ be a prime power and let $n$ be a positive integer. Consider monic polynomials of degree $n$ in $\mathbb{F}_q[x]$, whose coefficients include the constant term throug…

Let $f\in \mathbb Z[x]$ be an irreducible polynomial such that the integers in the set $f(\mathbb N)$ have no common factor other than $\pm 1$. Buniakowski's conjecture. The polyno…

Let $f_c(z)=z^2+c$. For positive integers $m,n$ with $m\mid n$, define the delta factors $\Delta_{n,m}$ by … and, when $m=n$, by … Here $\Phi^{\mathrm{cyc}}_k$ is the $k$-th cyclot…

Let $n\geq 7$ be odd, and let $G$ be the graph obtained from a path of length $n$ by attaching a pendant edge to some vertex of the path. Call $G$ irreducible when its adjacency ch…

Let $G$ be a finite simple tree, and call $G$ irreducible when its adjacency characteristic polynomial … is irreducible in $\mathbb{Q}[t]$. Akbari–Kumar–Mohar–Pragada conjecture. I…

Let $q$ be odd, let $n$ be a positive integer, and let $d\in\mathbb{F}_q^*$ be prescribed. Write $\chi_2$ for the quadratic character of $\mathbb{F}_q$, and let…

Let $\alpha,\beta\in\mathbb{F}_q$, let $\lambda\in\overline{\mathbb F}_p$, and let $f_\lambda(z)=z^d+\lambda$. Define…

Let $A\subset\mathbb Z$ satisfy $2\leq \#A<\infty$, and let $\mathcal P_n^{\mathrm{SRM},A}$ be the family of characteristic polynomials of symmetric $n\times n$ matrices with eleme…

Let $S=\{x^2+c_1,\dots,x^2+c_r\}$ for some distinct $c_1,\dots,c_r\in\mathbb{Q}$ and $r\geq2$. Let $M_S$ denote the semigroup generated by $S$ under composition, and call a subset…

Let $\Delta$ be one of the parameters considered in the paper, let $D=|\Delta|$ be prime, and let $h_\Delta$ denote the relevant degree of the trace polynomial $g_\Delta(x)$. The p…

Let $F_\Delta(x)$ be the generalized Fekete polynomial, and for each $n\geq 1$ let $r_\Delta(n)$ be the multiplicity of the cyclotomic polynomial $\Phi_n(x)$ in $F_\Delta(x)/x$, wi…

Hansen–Muller conjecture. Except for five small exceptional cases, for every $n$ and $q$ there exists a monic irreducible polynomial of degree $n$ over $\mathbb{F}_q$ with the pres…

Let $f(x)\in\mathbb{Z}[x]$ have degree $n\geq 1$. Define $L(f-g)=\sum_{j=0}^{n}|b_j-a_j|$ when $f(x)=\sum_{j=0}^{n}a_jx^j$ and $g(x)=\sum_{j=0}^{n}b_jx^j$. Turán's distance-two con…

Let $f$ be an irreducible polynomial over $\mathbb F_q$ of degree $n\ge 3$, and let $Q_c$ be a polynomial of degree $D$. Suppose that $n$ is odd and $D\equiv 2\pmod 4$, and that th…

Let $R$ be a polynomial coefficient ring, let $\boldsymbol{x}$ denote a tuple of variables, and write $R[\boldsymbol{x}]$ for the corresponding polynomial ring. For a nonconstant p…

Throughout, let $(f,g)$ be a pair of polynomials over $\mathbb{F}_q$ that are neither critical nor $p$-critical, with $f$ irreducible and $\deg g\geq 2$. Let $N_{f,g}(n)$ be the nu…

Throughout, let $(f,g)$ be a pair of polynomials over $\mathbb{F}_q$ that are neither critical nor $p$-critical, with $f$ irreducible and $deg g greater than or equal to 2$. Let…

Let $q=2^r$ with $r\geq 1$, let $n\geq 3$, and let $t_1,t_2,t_3\in\mathbb{F}_q$. For … where $\psi_q(n,t_1,t_2,t_3)$ is the von Mangoldt sum over monic degree-$n$ polynomials whose…

Let $q$ be a prime power, let $k$ and $l$ be positive integers, and write $\mathbb{F}_{q^k}$ for the degree-$k$ extension of $\mathbb{F}_q$. A polynomial…

Let $q$ be a prime power and let $n\geq 1$. Write … for the least power of $q$ that is at least $n$, and set … Let $T$ be the set of pairs $(g,h)\in\mathbb{F}_q[X]^2$ with…

Let $q$ be a prime power, let $n$ be a positive integer, and write $\bar{n}$ for the least power of $q$ that is at least $n$. For a polynomial $g\in\mathbb{F}_q[X]$, consider the p…

Prescribed-coefficients enumeration conjecture. There exist $\omega_1,\ldots,\omega_N\in\overline{\mathbb{Z}}$, all of norm $\sqrt q$, $\upsilon_1,\ldots,\upsilon_N\in\mathbb{Z}$,…

Let $q=2^r$, let $n\ge 3$, and let $t_1,t_2,t_3\in\mathbb{F}_q$. The function $f_q(n,t_1,t_2,t_3)$ counts the relevant objects with prescribed coefficients $t_1,t_2,t_3$. Period-24…

Let $q$ be a prime power, and let $k,l$ be positive integers. Write $c\mathcal{F}_k(q)$ for the set of elements of degree $k$ over $c\mathbb{F}_q$. Existence conjecture. There exis…