Characteristic variety

Remarks

Example: for the heat operator $P = \partial_t - \partial_x^2$ on $\R^2$ with variables $(t, x)$, the order is $2$ and $\sigma_2(P)((t, x), (\tau, \xi)) = -\xi^2$. So $\operatorname{Char}(P)$ consists of the covectors with $\xi = 0$, that is, $(\tau, 0)$ with $\tau \ne 0$. The lines $t = \text{const}$ have normal $(1, 0)$, so they are characteristic for the heat operator. Non-example: for the same operator the lines $x = \text{const}$ have normal $(0, 1)$, and $\sigma_2 = -1 \ne 0$ there, so they are noncharacteristic. The set $\operatorname{Char}(P)$ is conic: if $(x, \xi)$ lies in it, so does $(x, \lambda \xi)$ for every real $\lambda \ne 0$. The operator is elliptic exactly when $\operatorname{Char}(P)$ is empty.