11 problems

The weighted derivative-substitution conjecture. This mapping determines a map

The scalar strong BMV coefficient conjecture. Every $f_i$ belongs to $\mathbf{P}^{\mathrm{pos}}$, and consecutive coefficients satisfy

Let $A_i,B_i$ be positive definite matrices, let $g_i\in\mathbf{P}^{\mathrm{pos}}$, and let $f(x,y_1,\dots,y_e)\in\mathbf{P}_{1+e}$. The strong BMV conjecture. The determinant … ha…

Let $A$ and $B$ be positive definite matrices, and let $n$ be a positive integer. The BMV conjecture. The polynomial in $t$ … has all positive coefficients. The source presents thi…

Positivity and term count conjecture. The partition function $Z_n=Z_n(\alpha,\beta,q)$ is a positive polynomial with $(n-1)!$ terms, counted with multiplicity.

Let $n\geq 2$. The totally positive elements of the universal division algebra $\operatorname{USA}_n$ are those elements $a$ satisfying $a(X)\succeq 0$ for every…

Structure-coefficient positivity conjecture. For all $F,G,H\in L(M)$, $C_{FG}^H(q)\in\mathbb{N}[q]$. The conjecture was disproved by examples in rank $4$ and higher, including a un…

Coefficient-positivity conjecture. For every positive integer $n$ and every non-negative integer $k$ satisfying

Let $\psi_{\alpha}(x_1,\ldots,x_L)$ be the components of the polynomial solution of the qKZ equation of type A, evaluated in the limit $x_i\rightarrow 0$ for $i=1,\ldots,L$, and le…

Let $P_{3,6}$ be the cone of nonnegative ternary sextic forms, let $\Delta_{3,6}$ be its relevant cone of forms under consideration, and let $E(P_{3,6})$ denote the set of extremal…

Let $\Delta_{3,6}$ denote the cone of psd ternary sextic forms, and let an extremal form be an extremal element of this cone. A form has a zero at which it attains the relevant max…