= Solution
Write the order-$m$ <principal symbol> as $p_m(x,y;\xi,\eta)=\sum_{|\alpha|=m}a_\alpha(x,y)\xi^{\alpha_1}\eta^{\alpha_2}$. The conormal of the plane is $(1,0)$. Therefore the <non-characteristic hypersurface> hypothesis is exactly
$$
\boxed{a_{(m,0)}(0,y)=p_m(0,y;1,0)\ne0\quad\text{for }|y|<1.}
$$
In particular its value at the origin is nonzero, and continuity provides a uniform positive lower bound for its modulus in a sufficiently small neighborhood. No lower-order coefficient enters this condition.
Back to article page
