20 problems

Let $I$ be an interval. An irreducible polynomial $Q(x)=a_dx^d+\cdots+a_0\in\mathbb{Z}[x]$ with $a_d>0$, all roots in $I$, and $a_d^{-1/d}>t_{\mathbb{Z}}(I)$ is a nonmonic critical…

Let $I$ be an interval. Maximal obstruction existence conjecture. Every interval of length less than $4$ has a maximal obstruction. The paper proves existence when the interval has…

Let $I$ be an interval of length less than $4$. A maximal obstruction for $I$ is an obstruction whose value is maximal among the obstructions considered for $I$; denote its value b…

Let $I=[0,b]$ be an interval with $b e 0$ and $bleq 1$. Define … the smallest integer $n\geq 2$ for which $1/n\leq b$. Zero-endpoint interval conjecture. … The conjecture gives an…

Let $p$ be a prime with $p\geq3$, and define $c(n)$ by … Starting from the polynomial sequence associated with this recurrence, the prime-parameter conjecture. For $n\geq2$, the re…

Let $\textbf{\textsl{k}}=(k_1,\ldots,k_l)\in\mathbb{N}^l$ and let $Q_n^{\textbf{\textsl{k}}}(x,y)$ be the associated polynomials. Integrality and divisibility conjecture. The polyn…

Folklore conjecture. One has

For positive integers $m,n,H$, let … where $\mathrm{H}(h)$ is the maximum absolute value of the coefficients of $h$ and $L(h)$ is its leading coefficient. For coprime integer polyn…

For positive integers $m,n,H$, let … where $\mathrm{H}(h)$ is the maximum of the absolute values of the coefficients of $h$ and $L(h)$ is its leading coefficient. For coprime integ…

Let $f(x)\in\mathbb{Z}[x]$ have degree $n\geq 0$. A polynomial is squarefree if it is not divisible by the square of an irreducible polynomial over $\mathbb{Q}$. Define…

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 $R=\mathbb{Z}[\alpha,\beta]$, and define polynomials $\zeta_r(\alpha,\beta)$ recursively by … Writing $\beta=\alpha^2+8\varepsilon$, define … where $k!!=k(k-2)\cdots 3\cdot1$ f…

Square-free approximation conjecture. For any $f\in\mathbb{Z}[x]$ of degree $d$, there is a square-free polynomial $g\in\mathbb{Z}[x]$ of degree at most $d$ satisfying

Let $M_n(H)$ be the set of generalized degenerate monic integer polynomials of degree $n$ and height at most $H$. Growth conjecture. For every integer $n\ge 2$, … The preceding res…

Let $f(x)$ be a nonzero polynomial of integer coefficients. Let $Z_f:=\{x\in\mathbb{Z}:f(x)=0\}$, let $\{a_k\}_{k=1}^{\infty}:=\mathbb{Z}^+\setminus Z_f$ be arranged in increasing…

For an integer $n\ge 2$, let $A_n^*(1,H)$ denote the number of dominant integer polynomials of degree $n$ and height at most $H$, so that the total number of such polynomials is…

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 $g$ be a polynomial over the integers. For each integer $a$, define … A geometric progression of length $b$ is a sequence of $b$ terms with a common ratio. Geometric-progressio…

In the trigonometric case $y=0$, choose the singlet components to be coprime polynomials in $x$ and normalise their leading terms as specified in the source. Let $N$ be written as…

Arbitrarily large ratio conjecture. There exist factorizations with arbitrarily large ratios, even when restricted to palindromic $*$-symmetric factorizations of height-$1$ polynom…