12 problems

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

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 $S=\{x^2+c_1,\dots,x^2+c_r\}$ 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 $f(x)\in\mathbb{Z}[x]$ 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…

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 $q$ 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…

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 $q$ 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…

A monic cubic with depth-one emergent reducibility is a monic polynomial $f(x)\in\mathbb{Z}[x]$ of degree $3$ that is irreducible, while its first self-composition $f\circ f$ is re…

Let $\mathbb{F}_q$ be a finite field. An $F$-set is a set $\mathcal{S}\subset\mathbb{F}_q[x]$ such that, whenever $P\in\mathcal{S}$ is monic and irreducible with constant term…