33 problems

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…

Let $G$ be an absolutely almost simple group over a finitely generated field $k$, with characteristic of $k$ prime to the order of the Weyl group of $G$. For a field extension…

Le Bruyn's conjecture. The torus $T_n$ is never $k$-rational, except possibly when $n=6$.

Let $F$ be a field, let $\widetilde{F}/F$ be a field extension of degree $d$, and let $\varphi=N_{\widetilde{F}/F}$ be its norm form. For a field extension $K/F$, say that SNP hold…

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…

Stability of subrank under field extensions. There exists a polynomial $\psi$ of degree at most $d-1$ such that, for every field $\mathbb{K}$ and every tensor…

Let $K/F$ be a totally positive field extension of formally real fields, and let $A$ be a central simple algebra over $F$ of exponent $2$. The Pythagorean index conjecture asserts…

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 $F$ be a number field and $G$ a finite group. A finite extension $K/F$ is ramified over at most a single place when at most one place of $F$ ramifies in $K/F$. One-place ramifi…

Let $F$ be a global field and $G$ a finite group. For a finite extension $K/F$, let the ramification of $K/F$ mean the places of $F$ ramified in $K$, and consider the least possibl…

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…

Knaf's conjecture. If

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

Let $d>1$ and let ${\bf k}$ be a field with characteristic greater than $d$ or characteristic zero. For a polynomial $Q\in {\bf k}[x_1,\ldots,x_s]$ of degree $d$, write…

Let $B_1$ and $M_2(p)$ be as in the preceding construction, with $p_2$ the coefficient of $F^2$ in the cubic factor $P_3(F)$. Coefficient-field subextension conjecture. The field…

Let $B_1$ be the series under consideration and let $\mathbb{Q}(t,B_1)$ be the field extension generated by it over $\mathbb{Q}(t)$. Degree-four subfield conjecture. The extension…

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

Let $f_1,\ldots,f_m$ be the polynomials defining the square roots under consideration, and let rationalizability refer to the notion for sets of square roots of polynomials. Ration…

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 $L$.