  This is a the first in a series of two articles devoted to the question of
local solvability of doubly characteristic differential operators $L,$ defined,
say, in an open set $\Om\subset \RR^n.$
  Suppose the principal symbol $p_k$ of $L$ vanishes to second order at
$(x_0,\xi_0)\in T^*\Om\setminus 0,$ and denote by $Q_\H$ the Hessian form
associated to $p_k$ on $T_{(x_0,\xi_0)}T^*\Om.$ As the main result of this
paper, we show (under some rank conditions and some mild additional conditions)
that a necessary condition for local solvability of $L$ at $x_0$ is the
existence of some $\theta\in\RR$ such that $\Re (e^{i\theta}Q_\H)\ge 0.$
