62 problems

Schoenberg's conjecture. The quadratic inequality

Matsaev's conjecture. For every polynomial $p\in\mathbb{C}[z]$,

Let $d\ge 2$ be a natural number, let $P$ be a $d$-dimensional polytope in $\mathbb{R}^d$, and let $p_0(x),\ldots,p_{d-1}(x)$ be polynomials. Bosse–Grötschel–Henk conjecture. There…

The squared Bernstein-factor conjecture. The factor $\alpha(K,{\bf x})$ in this estimate should be replaced by $\alpha(K,{\bf x})^2$, so that

Let $E$ be a non-symmetric convex body in ${\bf R}^d$, let $\lambda(x)$ denote the density of its equilibrium measure in the interior, and let $\widetilde{G}(E,x)$ be the convex hu…

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 $f\in\mathbf{P}^{\mathrm{pos}}$ and define the Hankel matrix from its derivatives by … For a fixed size $d$, write $\det H((f(x),f'(x),f^{(2)}(x),\dots);d)$ for the correspondi…

Stability-index conjecture. Every $n$-dimensional polytope can be represented by $n$ polynomial inequalities.

Let $E\subset\mathbb{C}$ be a bounded non-degenerate continuum, and let $M_E$ denote the best constant associated with the norm inequality for polynomials. For the closed unit disk…

Agler–McCarthy's entropy conjecture. One has

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…

Hunter's conjecture. The global minimum of $h_{2k}$ on this normalized sphere is attained at $\tilde{a}$.

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

Let $\mathbb{T}$ be the unit circle, and for polynomials $P$ of degree at most $d$ let $\mathscr{C}_{d,p}$ be the smallest constant $C$ such that … where the $L^p(\mathbb{T})$ norm…

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

De Bruin–Sharma's conjecture. In the quartic case, the inequality

Let $d_N$ be the smallest constant for which the lower bound … holds in the factorization of a degree-$N$ polynomial into linear factors. Etayo's asymptotic conjecture. There is a…

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…

Erdős's conjecture. The Bernstein inequality can be strengthened to

Erdős's conjecture. If $f(z)$ does not vanish in $|z|<1$, then the factor $n$ in this upper bound can be replaced by $n/2$:

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…