17 problems

Let $f_1,\ldots,f_l,g\in\mathbb{C}\langle\underline{x}\rangle$. For a complex Hilbert space $\mathcal{H}$, let $\mathcal{B}(\mathcal{H})$ denote the bounded operators on…

Let $\mathbb{K}$ be a field, let $f_1,\ldots,f_l,g\in\mathbb{K}\langle\underline{x}\rangle$ be noncommutative polynomials, and for each $n\in\mathbb{N}$ and…

Kernel equality conjecture. For every such tuple,

Rank Nullstellensatz conjecture. The following are equivalent: (i) there is $K\in\mathbb{N}$ such that for every $n\in\mathbb{N}$ and every…

Makar-Limanov's conjecture.

Invariant-subspace certificate conjecture. The following are equivalent: (i) for every $n\in\mathbb{N}$ and every $\underline X\in\operatorname{M}_n(\mathbbm k)^d$, every joint inv…

L'vov–Kaplansky conjecture. If $f$ is multilinear, then its image in the matrix algebra $M_n(K)$ is a vector space for every $n\in\mathbb{N}$.

Let $K$ be an algebraically closed field, let $n\geq 2$ and $m\geq 1$ be integers, and let $T_n(K)$ denote the algebra of $n\times n$ upper triangular matrices over $K$. For…

Let $K$ be a field, let $n\geq 2$ be an integer, and let $f(x_1,\dots,x_m)$ be a multilinear polynomial. Say that $f$ satisfies $(S1)$ when it is a central polynomial, so its image…

Let $K$ be a field, let $n\geq 2$ and $m\geq 1$ be integers, and let $f(x_1,\dots,x_m)$ be a nonzero multilinear polynomial in $K\langle x_1,\dots,x_m\rangle$. Mesyan conjecture. I…

Let $\mathbb K\langle\underline{x}\rangle$ denote the free noncommutative polynomial algebra, let $\mathfrak r=(p_{ij})\in\mathbb K\langle\underline{x}\rangle^{L\times L}$ have non…

Eigenvector generation conjecture. For each $n\geq 1$, a generating set of $S_{\mathbb Q}\langle a,b\rangle_n$, consisting only of eigenvectors, may be obtained by applying all pos…

Kernel conjecture. For each $n\geq 1$,

Finite-type representing measure conjecture. The following are equivalent:

Let $f$ be a multilinear polynomial with complex coefficients, and let $d\in\mathbb{N}$. Consider the set of values obtained by evaluating $f$ on $M_d(\mathbb{C})$. Kaplansky–Lvov…

Helton–McCullough conjecture. If $p$ is irreducible and $\mathcal D_p$ is convex, then $\mathcal D_p$ is a ball.

Let $q$ be the partial Hessian of a symmetric polynomial, let $Z(0,A)$ be the associated middle matrix evaluated at $A\in\mathbb S_n(\mathbb R^g)$, let $\mu_\pm$ denote the number…