20 problems

Polynomial lower-bound conjecture. There exists $\eta\in[\mu,L]$ such that

Let $H^p$ denote the Hardy space on the unit disk, let $d54b$ be the unit circle, and let $f\in H^p$ satisfy $f(0)\neq 0$. An inner function is a bounded analytic function on the u…

Let $K$ be a compact subset of the complex plane, and write $K^\circ$ for its interior. A continuous function $f:K\to\mathbb C$ is zero-free on $K^\circ$ when $f(z)\neq 0$ for ever…

Let $K\subset {\mathbb{C}}\setminus \{0\}$ be a compact set. A polynomial of the indicated form is $\sum_{k=n/\tau_n}^{n}a_kz^k$, where $\tau_n\to 1$. Polar-set characterization co…

Let $K$ be a non-empty compact subset of the complex plane, and let $\delta_n$ and $C$ denote the quantities defined in the preceding results. Dilation asymptotic conjecture. … Thi…

Let $\Pi_d$ be the polynomial set associated with degree-$d$ approximation conditions, and let $\overline{\Pi^*_{2^7}}$ denote the closure of the polynomial set obtainable with sev…

Let $\Pi_d$ be the polynomial set associated with degree-$d$ approximation conditions, and let $\overline{\Pi^*_{2^6}}$ denote the closure of the polynomial set obtainable with six…

Let $\boldsymbol{\Omega}$ be the compact support of a positive Borel probability measure $\mu$ on $\mathbb{R}^d$, and let $f$ be its positive, continuous density with respect to Le…

Let $1<p<\infty$, $f\in H^p$, and $n\in\mathbb{N}$. If $f(0)\neq 0$, then the optimal polynomial approximant $q_{n,p}[f,1]$ is the polynomial of degree at most $n$ that best approx…

Let $K$ be a compact set with connected complement, and let $f$ be a continuous function on $K$ that is analytic in the interior $K^\circ$ and zero-free on $K^\circ$. Lavrentiev–An…

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…

For $T>1$, let $A$ be the open arc of the complex unit circle … and define … For $F\in C^1(A)$, write $F'(z)$ for its derivative along the arc. Jackson inequality for polynomials o…

Width-dependent Turán conjecture. There exists an absolute constant $c>0$ such that, for every compact convex domain $K\Subset\mathbb C$ and every $p\in\mathcal P_n(K)$,

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 $E$ be a regular compact set in $\mathbb{C}$, let $(a_n)_{n=0}^{\infty}$ be a pseudo Leja sequence of bounded Edrei growth in $\partial \widehat{E}$, and let…

Optimality conjecture. There exists an integer triple $(i,j,k)$, not all zero, with $i+j+k\leq 2n$, such that at least one of

Let ${\bf c}^*=(\beta,0)$ be as in the proof of Theorem. For $L>0$, define … Let $S_1$ be any disk in the connected component of $D_{\beta,L}$ containing the origin, and let $S_2$…

Let $n\geqslant 2$ be an integer, let $0<\mu<\frac{n+1}3$, and let $w>n-2\mu$. Define … where $H(P)$ is the absolute height of $P$, and let $\dim$ denote Hausdorff dimension. Baker…

Let $\alpha\in\mathbb{N}$, and let $f\in {C^\alpha}^*[0,1]$ be a function bounded strictly between $0$ and $1$. Here ${C^\alpha}^*[0,1]$ consists of functions that are…