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  "body": "\n## Definition\n\nLet $P$ be a [[linear differential operator|6DZETZ5T9G]] of order $m$ on an open set $\\Omega \\subseteq \\R^n$, with [[principal symbol|F31T650DKY]] $\\sigma_m(P)$.\nThe *characteristic variety* of $P$ is the set\n$$\\operatorname{Char}(P) = \\set{(x, \\xi) \\in \\Omega \\times (\\R^n \\setminus \\set{0}) : \\sigma_m(P)(x, \\xi) = 0}.$$\nA nonzero vector $\\xi$ is *characteristic* for $P$ at $x$ if $(x, \\xi) \\in \\operatorname{Char}(P)$.\nA $C^1$ hypersurface $S \\subseteq \\Omega$ is *noncharacteristic* at a point $x \\in S$ if a [[normal vector|0WQJJCCTC8]] $\\nu$ of $S$ at $x$ satisfies $\\sigma_m(P)(x, \\nu) \\ne 0$.\n\n## Remarks\n\nExample: for the heat operator $P = \\partial_t - \\partial_x^2$ on $\\R^2$ with variables $(t, x)$, the order is $2$ and $\\sigma_2(P)((t, x), (\\tau, \\xi)) = -\\xi^2$.\nSo $\\operatorname{Char}(P)$ consists of the covectors with $\\xi = 0$, that is, $(\\tau, 0)$ with $\\tau \\ne 0$.\nThe lines $t = \\text{const}$ have normal $(1, 0)$, so they are characteristic for the heat operator.\nNon-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.\nThe set $\\operatorname{Char}(P)$ is conic: if $(x, \\xi)$ lies in it, so does $(x, \\lambda \\xi)$ for every real $\\lambda \\ne 0$.\nThe operator is [[elliptic|B4TTNHSTRX]] exactly when $\\operatorname{Char}(P)$ is empty.\n\n## Explanations\n\n### L1 human\n\nThe characteristic variety is the zero set of the principal symbol, with the zero covector removed.\n",
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