Characteristic hypersurface

A hypersurface $\Sigma =\{\varphi =0\}$ is characteristic for a differential equation at $x\in \Sigma$ when its principal symbol vanishes on the conormal $d\varphi (x)$:It is non-characteristic where this quantity is nonzero.

A non-characteristic hypersurface satisfies $p(x,d\varphi (x))\ne 0$. This lets the equation solve for the highest derivative normal to the hypersurface.