4 problems

Rationality conjecture. If the polynomials $\varphi_i$ have rational coefficients, then

Let $K$ be a topological local field over $\boldsymbol{k}$, and let $\operatorname{Res}^{\mathrm{BT}}_{K / \boldsymbol{k}}:\Omega^n_{K / \boldsymbol{k}}\to\boldsymbol{k}$ be its Be…

Let $K$ be an $n$-dimensional topological local field over $\boldsymbol{k}$. Write $obreakOmega^n{K / boldsymbol{k}}twoheadrightarrowOmega^{n,mathrm{sep}}{K / boldsymbol{k}}$ for…

Let $J\in \hat{\mathcal{B}}_1^r({\mathbb{R}}^2)$ be supported in a bounded open set $U\subset {\mathbb{C}}$, with $\operatorname{Ind}_J(w)=0$ for all $w\in U^c$, and suppose there…