17 problems

Let $\mathbb{K}$ be a field with … and let $P\in\mathbb{K}[x_1,\dots,x_n]_d$ be a homogeneous polynomial of degree $d$. Here $\operatorname{str}_{\mathbb{K}}(P)$ denotes the streng…

Let $\mathbb{K}$ be a field, let $d\geq 2$ be an integer, and let … Here $\operatorname{PR}_{\mathbb{K}}(f)$ denotes the partition rank of $f$, and $\overline{\mathbb{K}}$ is an al…

Let $d\geq 2$, let $\mathbb{K}$ be a field, let $\mathsf{W}\subseteq\mathbb{K}^{n_1}\otimes\cdots\otimes\mathbb{K}^{n_d}$ be a linear subspace, and let $k$ be a positive integer sa…

Let $d\geq 2$, let $\mathbb{K}$ be a field, and let $\mathsf{W}\subseteq\mathbb{K}^{n_1}\otimes\cdots\otimes\mathbb{K}^{n_d}$ be a linear subspace. Write $\overline{\mathbb{K}}$ fo…

Let $d\geq 2$, let $q$ be a prime power, and let $T\in\mathbb{F}_q^{n_1}\otimes\cdots\otimes\mathbb{F}_q^{n_d}$. Write $\overline{\mathbb{F}}_q$ for an algebraic closure of…

Let $K$ be a field, let $\bar{K}$ be an algebraic closure of $K$, and let $X_K\subset K^n$ be a nonsingular $K$-rational algebraic variety. Assume that…

Let $L/K$ be a separable field extension of squarefree degree. Say that $L/K$ has the parallel no-HGS property if it admits a Hopf–Galois structure while a parallel extension assoc…

Chowla matching conjecture. If $B$ is a Chowla subspace, then $A$ is matched to $B$.

Linear product-set conjecture. Then every pair of $K$-subspaces $S\subset A$ and $T\subset B$ satisfies

Let $k$ be a complete metrized field, let $X$ be a projective variety defined over $k$, and let $f\colon X\dashrightarrow X$ be a dominant rational map. For any complete metrized f…

The dimension formula for the largest matchable subspace. If $\langle AB\rangle\neq A$ and $A$ is not matched to $B$, then

Let $\mathbf k$ be a field, let $\bar{\mathbf k}$ be its algebraic closure, let $V_1,\ldots,V_d$ be $\mathbf k$-vector spaces, and let $L\subset (V_1\otimes\cdots\otimes V_d)^\vee$…

Let $\mathbf k$ be a field, let $\bar{\mathbf k}$ be its algebraic closure, and let $P$ be a multilinear $\mathbf k$-polynomial of degree $d\geq 2$. Write $r_{\mathbf k}(P)$ for it…

Arithmetic Persistence Conjecture. The base change $X_L$ is arithmetically hyperbolic over $L$.

Let $(K,\nu)\subset (L,\omega)$ be in situation, with valuation rings $\mathcal{O}_\nu$ and $\mathcal{O}_\omega$. Assume in addition that the field extension $L|K$ is separable. Se…

Let $a>b\geq 0$ be integers and let $[Z_1:Z_2:Z_3]\in \mathbb{P}^2(\mathbb{Q})$. Brassil–Reichstein's conjecture. The system … … has no solution $(a,b,[Z_1:Z_2:Z_3])$. This conject…

Fourth-layer compositum conjecture.