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  "body": "\n## Statement\n\nLet $P(z) = a_n z^n + \\cdots + a_1 z + a_0$ be a polynomial of degree $n \\ge 1$ with real coefficients and $a_n \u003e 0$.\nBuild rows of numbers as follows.\nThe first row is $(a_0, a_1, \\dots, a_n)$.\nIf a row $(x_0, x_1, \\dots, x_m)$ has $m \\ge 3$, the next row is $(y_0, \\dots, y_{m-1})$ with\n$$y_k = x_0 x_k - x_m x_{m-k}, \\qquad k = 0, 1, \\dots, m-1.$$\nStop at the first row that has exactly $3$ entries; for $n \\le 2$ only the first row is built.\nThen $P$ is [[Schur stable|X11Q0G8PNA]] if and only if all of the following hold.\n\n1. $P(1) \u003e 0$.\n2. $(-1)^n P(-1) \u003e 0$.\n3. $\\abs{a_0} \u003c a_n$.\n4. Every row $(y_0, \\dots, y_{m-1})$ built after the first satisfies $\\abs{y_0} \u003e \\abs{y_{m-1}}$.\n\n## Remarks\n\nThe proof rests on the Schur–Cohn reduction.\nWrite $P^*(z) = z^n P(1/z)$, so that $\\abs{P^*(z)} = \\abs{P(z)}$ on the unit circle because the coefficients are real.\nWhen $\\abs{a_0} \u003c a_n$, [[Rouché's theorem|0D430QR02W]] shows that $P$ is Schur stable if and only if the polynomial $(a_n P(z) - a_0 P^*(z))/z$ of degree $n - 1$ is Schur stable.\nThe next row of the table lists the coefficients of this reduced polynomial in reverse order and with the opposite sign, so the table repeats the reduction until degree $2$.\nThe conditions 1 and 2 are necessary, since a real polynomial with $a_n \u003e 0$ and no roots in $[1, \\infty)$ or $(-\\infty, -1]$ has the stated signs at $\\pm 1$.\nFor $n = 2$ the criterion reads $\\abs{a_0} \u003c a_2$ and $\\abs{a_1} \u003c a_0 + a_2$.\nFor instance $z^2 - z + \\tfrac{1}{2}$ passes: $\\tfrac{1}{2} \u003c 1$ and $1 \u003c \\tfrac{3}{2}$.\nThe polynomial $z^2 - \\tfrac{5}{2} z + 1 = (z - 2)(z - \\tfrac{1}{2})$ fails condition 1, since $P(1) = -\\tfrac{1}{2}$.\n\n## Explanations\n\n### L1 human\n\nThe criterion decides whether all roots of a real polynomial lie inside the unit circle without finding the roots.\nIt checks a few signs and inequalities that come from simple products of the coefficients.\nEngineers use it to see whether a digital filter or a sampled control loop is stable.\n\n### L2 human\n\nThe criterion is the discrete-time analogue of the [[Routh–Hurwitz stability criterion|KJF8Y19BNV]], which tests whether roots lie in the left half-plane.\nCombined with the stability theorem for linear recurrences, it decides from the coefficients whether every solution of a linear recurrence tends to zero.\nA degree $n$ polynomial needs $n + 1$ conditions in total: three simple ones and $n - 2$ from the table.\nWhen a parameter enters the coefficients, the conditions give the region of parameters where the system is stable.\n",
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