11 problems

Let $P(x)$ be a polynomial, and let $\bar{\mathbf{d}}$ denote upper density. For a positive integer $T$, consider the set of integers $n$ for which $p-1$ divides $P(n)$ for some pr…

Let $F(x)\in\mathbb{Z}[x]$, and let $\Psi_F^*(x,y)$ denote the counting function for the relevant integers whose values under $F$ satisfy the smoothness condition encoded by the so…

Verstraëte's dichotomy conjecture. For some constant $\rho=\rho(k,f)$ depending only on $k$ and $f$, the maximal size of such a set $A$ is either

Let $q=p^k$ be a prime power, and let $f\in\mathbb{F}_q[T][X]$ be a fixed irreducible polynomial with $X$-degree $d\geq 2$. Define … Let $V_f:=\{g\in\mathbb{F}_q[T]:f(X+g)=f(X)\}$…

Let $f\in\mathbb{Z}[x_1,\ldots,x_h]$ be a primitive squarefree polynomial, let $\mathcal{A}$ be an arithmetic progression, and let $\mathcal{A}(X)$ denote its points of norm at mos…

Let $f\in\mathbb{Z}[x_1,\ldots,x_h]$ be primitive and squarefree, and let $\mathbb{Z}^h(X)=\{{\rm v}\in\mathbb{Z}^h:|{\rm v}|\leq X\}$. Squarefree conjecture, alternative version.…

Let $h\in\{1,2\}$, let $f\in\mathbb{Z}[x_1,\ldots,x_h]$ be primitive and squarefree, and let $\mathbb{Z}^h(X)=\{{\rm v}\in\mathbb{Z}^h:|{\rm v}|\leq X\}$. Define…

For a polynomial $P\in{\mathbb Z}[x]$, let $\Psi_P(x,y)$ count the integers among $\{P(n):1\leq n<x\}$ whose prime factors are all at most $y$. Weak smooth-values conjecture. For e…

Let $P\in{\mathbb Z}[x]$ be an irreducible polynomial of degree $d$, and let $\Psi_P(x,y)$ be the number of integers in the set $\{P(n):1\leq n<x\}$ whose prime factors are all at…

Squarefree-value residue class conjecture. For all but finitely many primes $p$, the set $\mathcal S(f)$ contains infinitely many elements from every nonzero residue class modulo…

Let $f_1(x),\ldots,f_r(x)\in\mathbb{Z}[x]$ and let $f(x)=f_1(x)\cdots f_r(x)$. Assume that the $f_i$ are distinct and irreducible in $\mathbb{Z}[x]$, have positive leading coeffici…