9 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…

Let $k\subseteq K$ be fields, let $R=k[X_1,\ldots,X_n]$, and let $I$ be an ideal of $R$. For $1\leq i\leq 4$, define … Write $\mathsf{Z}_K(I)$ for the common zero set of $I$ over…

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…

Gori–Sarfatti–Vlacci conjecture. For every $n$ and every left ideal $I\subseteq R$, one has

Let $I(\text{\eufmh})$ be the algebra specified in Theorem, and let $m$ be a maximal ideal of $I(\text{\eufmh})$. Conjecture 13.5.1. There is a $\lambda\in\text{\eufmh}^*$ such tha…

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…

Degree-cutter equations conjecture. These equations may decrease the minimal degree of the Nullstellensatz certificate, if one exists. The conjecture proposes that adding equations…