23 problems

Let $f\in\mathbb{Z}[x]$ be irreducible and have trivial fixed divisor. Bunyakowski's conjecture. The value $|f(n)|$ should be prime for infinitely many $n\in\mathbb{Z}$. This is a…

Let $m$ be fixed, let $f(x)=f_0+\cdots+f_mx^m$, and let $g(x)=g_0+\cdots+g_nx^n$ be a degree $n$ polynomial. Here $H$ denotes the height of an integer polynomial or integer, and…

Let $f(t)$ be an irreducible polynomial of degree at least $4$. For $N>0$, consider integers $t\in[N,2N]$ for which $f(t)$ is divisible by the square of a prime larger than…

Permissible-polynomial conjecture. Every permissible polynomial is a Chinburg polynomial, with the displayed linear combination determined by its $(f_1,\ldots,f_s)$-permissibility.…

Let … be a polynomial with fixed outer coefficients $a_n=1$ and $a_0=b$, and let $E_{n,b}(H)$ denote the number of such polynomials of height at most $H$ whose Galois group is not…

Schinzel–Tijdeman conjecture. The equation has only finitely many solutions $(x,y,z)\in\mathbb{Z}^3$ with $yz\ne0$.

Let $q$ be a fixed power of an odd prime and let $c>0$ be fixed. For any polynomial $F\in\mathbb{F}_q[x][T]$ that is separable in the variable $T$, define the Liouville function by…

Let … be the degree-six truncation of the binomial expansion, and let $\mathfrak{S}_6$ denote the full symmetric group on six symbols. Filaseta–Moy conjecture. The Galois group of…

Let $\Delta\in\{4p,-4p,3p,-3p\}$, let $f_\Delta(x)$ be the associated Fekete polynomial, and let $g_\Delta(x)$ be its trace polynomial. Let $h_\Delta$ be the degree of $g_\Delta$,…

For $m\geq 2$ and $n\geq 1$, let $P_{m,n}$ denote the multiplier polynomial in the quadratic case $d=2$. A polynomial is called 2-special when it has the special coefficient and di…

Let $f\in\mathbb{Z}[x]$ be an irreducible polynomial. A large prime factor witness (LPFW) for $f$, a root bound $\rho$, and a factor degree lower bound $\Delta$ is an integer $n$ s…

For fixed integers $n,N\geq2$, let … and let $K_f\subset\mathbb C$ be the splitting field of $f$ over $\mathbb Q$. Write $\eta_{n,N}$ for the proportion of polynomials in $E_n(N)$…

Let $(m(F_n))_{n\in\mathbb N}$ be a sequence with $F_n\in\mathcal L$, where the $F_n$ are irreducible integer polynomials and $\operatorname{dim}(F_n)\to\infty$ as $n\to\infty$. Fo…

Let $q$ be a prime power. An $F$-set is a subset of the monic irreducible polynomials in $\mathbb{F}_q[x]$ closed under taking monic irreducible divisors of $f(x)-f(0)$ for each me…

Throughout, let $q$ be a prime power, let $I_q$ be the set of all monic irreducible polynomials in $\mathbb{F}_q[x]$, and call a subset $A$ of $I_q$ an $F$-set if, for every…

Let $B^{\circ}_n(q)$ be the polynomial defined by … Let $v_n$ be the valuation of $B^{\circ}_n(q)$, namely the maximal integer $r$ such that $q^r$ divides $B^{\circ}_n(q)$. Valuati…

Let $F_1,\ldots,F_m\in\mathbb{F}_q[t][x]$ satisfy the hypotheses of Theorem 1, except that the $F_i$ need not be monic in $x$ and the conditions $n\geq \operatorname{sl} F_i$ are o…

Let $F_1,\ldots,F_m\in\mathbb{F}_q[t][x]$ be non-associate irreducible polynomials that are separable over $\mathbb{F}_q(t)$, and let $n\geq 3$. The monicity condition on the $F_i$…

Consider polynomials with integer coefficients and $l_1$ norm $2n$, and let their vanishing at $1$ mean the largest integer $k$ such that $(x-1)^k$ divides the polynomial. Prouhet–…

Let ${\mathbb F_q}[t]$ be the polynomial ring over the finite field ${\mathbb F_q}$. For a set ${\mathcal H}\subset {\mathbb F_q}[t]\setminus\{0\}$, being van der Corput means that…

Birch–Chowla–Hall–Schinzel conjecture. For every such pair, one has

Prime-polynomial conjecture. In case (i), $u+v\not\equiv0\pmod p$, infinitely many primes $q$ with $\kappa\not\equiv0\pmod p$ make $A_p(\rho)X^p+\cdots+A_0(\rho)$ reducible modulo…

Let $f_0,\ldots,f_n$ be one-variable polynomials over a field of characteristic zero, satisfying the hypotheses under which inequality (BB) is stated, and assume that there are no…