It is shown that the system of two additive equations a_1 x_1^k + \ldots + a_s x_s^k = b_1 x_1^k + \ldots + b_s x_s^k =0 where $k \ge 2$ and $a_j$, $b_j$ are any given integers, has non-trivial solutions in all $p$-adic fields provided only that $s > 8k^2$. The constant 8 can be reduced when $k$ is not a power of 2. It is expected, in accordance with a classical conjecture of Artin, that the bound $8k^2$ can be replaced by $2k^2$.

2000 Mathematical Subject Classification:11D72.