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  "body": "\n## Definition\n\nLet $P(z) = a_n z^n + a_{n-1} z^{n-1} + \\cdots + a_0$ be a polynomial of degree $n \\ge 1$ with complex coefficients.\nThe polynomial $P$ is *Schur stable* if every complex [[root|019MA1VTR2]] $\\lambda$ of $P$ satisfies $\\abs{\\lambda} \u003c 1$.\n\n## Remarks\n\nExample: $P(z) = z^2 - z + \\tfrac{1}{2}$ has roots $\\tfrac{1}{2} \\pm \\tfrac{1}{2} i$.\nEach has modulus $\\sqrt{1/4 + 1/4} = 1/\\sqrt{2} \u003c 1$, so $P$ is Schur stable.\n\nNon-example: $P(z) = z^2 - 1$ has roots $1$ and $-1$.\nThey have modulus exactly $1$, so the strict inequality fails and $P$ is not Schur stable.\n\nSome authors call such polynomials simply *stable* in a discrete-time context, or say that $P$ has all its zeros in the open unit disc.\nThe polynomial $P$ is Schur stable if and only if its reversed polynomial $z^n P(1/z)$ has all its roots outside the closed unit disc, provided $a_0 \\ne 0$.\n",
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