9 problems

Pollio–Rapinchuk's conjecture. The multinorm principle for $L=L_1\oplus L_2$ holds whenever every subfield of $L_1\cap L_2$ satisfies the Hasse Norm Principle. The conjecture was r…

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 $F$ be a nondegenerate norm form, and let $V(F)$ denote the volume of the region in $\mathbb{R}^n$ defined by the corresponding equation (1), with the dimension and degree as i…

Let $F$ be a nondegenerate norm form of degree $d$ in $n$ variables, and let $N_F(m)$ denote the number of integral solutions to the corresponding equation $F(X_1,\ldots,X_n)=m$. T…

Let $P_1,\ldots,P_n\in\mathbb{Q}[t]$ be pairwise distinct irreducible monic polynomials. Let $k_i=\mathbb{Q}[t]/(P_i(t))$ be the corresponding number fields, and let $a_i\in k_i$ d…

Let $C^{\operatorname{mod}\,2}_{n,D}(e_1,e_2,t_1,t_2,M)$ count primes $a^n+Db^n<M$ with $F^{-1}(t_1)<b/a<F^{-1}(t_2)$ and $a\equiv e_1\pmod 2$, $b\equiv e_2\pmod 2$. If the two rel…

Cusick's conjecture. For $n\geq 3$, if $\lambda(B_n)>0$, then $B_n$ is a norm form.

Let $k$ be a number field and let $K/k$ be a cyclic extension of degree $3$. For a polynomial $f\in k[t]$ of degree $6$, let $\mathcal{S}_{f}$ denote the variety defined by the nor…

Let $K$ be a number field embedded in ${\mathbb R}^n$, and let $\overline{K}={\mathbb R}^n$ be the topological closure of its image under the standard embedding. Let $M_j(K)$ and…