The local-field fewnomial root-count conjecture

Let $Omega_L(n,k)$ denote the maximum number of non-degenerate roots in a local field $L$ of a system of $n$ polynomials in $n$ variables having at most $k$ monomials in total. There are absolute constants $C_2>C_1>0$ such that, for any local field $L$ of characteristic $0$ and any $n,k\geq 2$,

$$(n+k-1)^{C_1\min\{n,k-1\}}\leq\Omega_L(n,k)\leq(n+k-1)^{C_2\min\{n,k-1\}}.$$

This conjecture predicts matching polynomial-order bounds for fewnomial systems over all characteristic-zero local fields; the preceding discussion gives substantially different known bounds in Archimedean and non-Archimedean settings, while the proposed uniform estimates remain open.