23 problems

Let $K$ be a convex body in a normed space, let $x\in\operatorname{int}K$, and let $B_n(K,x)$ denote the Bernstein factor at $x$ for polynomials of degree at most $n$. Write $w(K)$…

Let $c_n(X)$ denote the $n$th linear polarization constant of a normed space $X$, and let ${\bf R}^n$ be the real Euclidean space. Révész–Sarantopoulos conjecture. … This conjectur…

Let $K\subset\mathbb{C}$ be compact and connected with more than one point. Let $M_K$ be the smallest number such that, whenever $p$ has degree $n\geq1$ and $p=p_1\cdots p_m$, … He…

Let $d\in\mathbb{N}$. Let $H_n$ be the hyperoctahedral group acting on $\mathbb{R}^n$ by coordinate permutations and sign changes. A measure $\mu$ on $\mathbb{R}^n$ is isotropic if…

Let $\sigma_2(n):=\sum_{d\mid n}d^2$, and let $\Delta_n^{\sigma_2}(x)$ denote the difference associated with the D'Arcais polynomials for $g(n)=\sigma_2(n)$. Plane-partition positi…

Carbery–Wright conjecture. There exists an absolute constant $C>0$ such that

The linear upper-bound conjecture. If $d\geq 1$ and $p\geq 2$, then

Let $x_1\leq\ldots\leq x_k$ be a chain, let $P_{\mathrm{KS}}$ be its Kahn–Saks polynomial, and let $N(P_{\mathrm{KS}})$ denote its normalization. For exponent vectors $\alpha$ and…

Let $p$ be a polynomial with nonnegative coefficients, and let $\mathscr{W}(p)$ denote the polynomial obtained by the operator $\mathscr{W}$. Fischer–Kubitzke conjecture. For all s…

Let $K\Subset\mathbb C$ be a compact convex domain with width $w$ and diameter $d$, and let $\mathcal P_n(K)$ denote the polynomials of degree at most $n$ whose zeros lie in $K$. F…

Let $K\Subset\mathbb C$ be a compact convex domain, let $0<q<\infty$, and let $\mathcal P_n(K)$ denote the polynomials of degree at most $n$ whose zeros lie in $K$. For a polynomia…

Permanental Sihong polynomial conjecture. The polynomial above is totally nonnegative for all $n$. This would provide a permanental analogue of the known totally nonnegative determ…

Let … where $c\in\mathbb{C}$, $m\in\mathbb{N}$, $s_k\geq 1$, $ms_k\in\mathbb{N}$, and $|a_k|\geq 1$. For any branch $f(z)=P(z)^{1/m}$, write $\|\cdot\|_{\mathbb{T}}$ for the maximu…

Let $\mathcal{A}$ be a C-algebra, let $d\in\mathbb{N}\setminus\{1\}$, and let … be a polynomial over $\mathcal{A}$ with $a_1,\ldots,a_d\in\mathcal{A}$. Suppose that … on…

Let $\mathcal{A}$ be a C-algebra, let $d\in\mathbb{N}\setminus\{1\}$, and let … be a polynomial over $\mathcal{A}$ with $a_1,\ldots,a_d\in\mathcal{A}$. Suppose that its formal deri…

Let $P_n(x)$ be the polynomials associated with generalized plane partitions, and define … Let $a$ and $b$ be integers satisfying $a-1>b\geq 0$ and $(a,b)\notin\{(4,2),(6,4)\}$. Ge…

Let $P_n(x)$ denote the polynomials associated with the generalized plane partitions, and let $a$ be an integer with $a\geq 12$. Turán inequality conjecture. For every real number…

Let $Q(z)=1+z^n$, where $n$ is a positive integer, and let $\|\cdot\|_s$ denote the corresponding quasi-norm on the unit circle. Baernstein's conjecture. For every…

Jensen-polynomial zero-bound conjecture. For every such $p$ and every relevant index $i$,

Even-degree weighted Hawaiian conjecture.

Weighted Hawaiian conjecture.

Forsgård–Shapiro coefficient-difference conjecture. The number of real zeros of $f$ does not exceed $v(f)$.

Let $R_d(x)$ and $P_d(x)$ be the polynomials whose ratio is under consideration. Lower-bound conjecture. If $d\ge5$ and $x\ge\frac12(d+\sqrt{d+2})$, then … This conjecture is prese…