4 problems

Generalized Vinogradov mean value conjecture. For every $\varepsilon>0$,

Let $k\in\mathbb{N}$. For each positive integer-valued function $C:\mathbb{N}\rightarrow\mathbb{N}$, let $P_C(k)=\sup_{l\geq 2}\{S_C(k,l)\}$, where $S_C(k,l)$ is the minimum number…

Artin's conjecture. Such a system has a non-trivial $p$-adic solution for every prime $p$ whenever

Let $S$, $R$, and $k_i$ be natural numbers, and let $a_{ij}$ be integers. Consider the system of diagonal equations … A solution is non-trivial when $(x_1,\ldots,x_S)\ne\mathbf{0}$…