Open-problem statements, with their sources attached.

Is there some constant $c > 0$ such that, for all large enough $n$ and all polynomials $P$ of degree $n$ with coefficients in $\{-1, 1\}$, $\max_{|z|=1} |P(z)| > (1 + c) \sqrt{n}?$

Is there a polynomial $f:\mathbb{Z}\to \mathbb{Z}$ of degree at least $2$ and a set $A\subset \mathbb{Z}$ such that for any $z\in \mathbb{Z}$ there is exactly one $a\in A$ and $b\in \{ f(n) : n\in\mathbb{Z}\}$ such that $z=a+b$?

Probably there is no such $A$ for the polynomial $X^3$.

Probably there is no such $A$ for the polynomial $X^k$ for any $k \ge 2$. This is asked in [Sek59].

Let $f(z)=\sum_{0\leq k\leq n} \epsilon_k z^k$ be a random polynomial, where $\epsilon_k\in \{-1,1\}$ independently uniformly at random for $0\leq k\leq n$.

Is it true that, if $R_n$ is the number of roots of $f(z)$ in $\{ z\in \mathbb{C} : \lvert z\rvert \leq 1\}$, then

almost surely?

There is some ambiguity as to whether the intended coefficient set is $\{-1, 1\}$ or $\{0, 1\}$, see erdos_522.variants.zero_one for the alternate version.

Let $f(z)=\sum_{0\leq k\leq n} \epsilon_k z^k$ be a random polynomial, where $\epsilon_k\in \{0,1\}$ independently uniformly at random for $0\leq k\leq n$.

Is it true that, if $R_n$ is the number of roots of $f(z)$ in $\{ z\in \mathbb{C} : \lvert z\rvert \leq 1\}$, then

almost surely?

Is there any polynomial $f(x, y) \in \mathbb{Q}[x, y]$ such that $f : \mathbb{Q} \times \mathbb{Q} \rightarrow \mathbb{Q}$ is a bijection?

Let $P(x), Q(x) ∈ ℝ[x]$ be two monic polynomials with non-negative coefficients. If $R(x) = P(x)Q(x)$ is a $0,1$ polynomial (coefficients only from $\{0,1\}$), then $P(x)$ and $Q(x)$ are also $0, 1$ polynomials.

The Casas-Alvero conjecture states that in characteristic zero, if a monic polynomial P has the Casas-Alvero property, then P = (X - α)ᵈ for some α.

The second Cuboid conjecture

The third Cuboid conjecture

The Inverse Galois Problem: every finite group is isomorphic to the Galois group of a Galois extension of the rationals.

Given a complex polynomial $p$ of degree $d ≥ 2$ and a complex number $z$ there is a critical point $c$ of $p$, such that $|p(z)-p(c)|/|z-c| ≤ |p'(z)|$.

Sendov's conjecture* states that for a polynomial $f(z)=(z-r_{1})\cdots (z-r_{n}),\qquad (n\geq 2)$ with all roots $r_1, ..., r_n$ inside the closed unit disk $|z| ≤ 1$, each of the $n$ roots is at a distance no more than $1$ from at least one critical point.

The Bateman-Horn Conjecture* Given a finite collection of distinct irreducible polynomials non-constant $f_1, f_2, \dots, f_k \in \mathbb{Z}[x]$ with positive leading coefficients that satisfy the Schinzel condition, the number of positive integers n ≤ x for which all polynomials $f_i$ are simultaneously prime is asymptotic to: $C(f_1, f_2, \dots, f_k) x / (log x)^k$ where $C$ is the Bateman-Horn constant given by the convergent infinite product: $C = \frac{1}{D}\prod_{p\in\mathbb{P}} (1 - 1/p)^(-k) · (1 - \omega_p/p)$ Here $\omega_p/p$ is the number of residue classes modulo $p$ for which at least one polynomial vanishes.

The Schinzel condition ensures that for each prime $p$, there exists some integer $n$ such that $p$ does not divide the product $f_(n) f_2(n) \dotsb f_(n)$, which guarantees the infinite product converges to a positive value.