5 problems

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…

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

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…

Asymptotic certificate-degree conjecture. For every field $\mathbb K$ and every positive integer $m$, there exists a constant $k_0$ with the following property: for each $k>k_0$ an…

Let $F=\{f_1,\ldots,f_k\}$ be a system of tropical polynomials in $n$ variables, and let $M_N$ be the finite Macaulay matrix obtained from the monomial multiples of the $f_i$ of de…