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  "body": "\n## Definition\n\nLet $n \\ge 2$ be an integer and let $\\sigma$ be a real number.\nLet $\\mathcal{Z}_{n,\\sigma}$ be the set of real polynomials of the form\n$$p(x) = x^n + \\sigma x^{n-1} + q(x),$$\nwhere $q$ is any real polynomial of [[degree|0DPF6WRZW6]] at most $n - 2$.\nThe *Zolotarev polynomial* $Z_{n,\\sigma}$ is the element of $\\mathcal{Z}_{n,\\sigma}$ with the least value of $\\max_{-1 \\le x \\le 1} \\abs{p(x)}$.\n\n## Remarks\n\nThe minimiser exists and is unique.\nIndeed $Z_{n,\\sigma} = g - q^*$, where $g(x) = x^n + \\sigma x^{n-1}$ and $q^*$ is the [[best approximation|7N4988HN3P]] of $g$ in the uniform norm on $[-1, 1]$ by polynomials of degree at most $n - 2$.\n\nExample: for $\\sigma = 0$ the Zolotarev polynomial is $2^{1-n} T_n$, where $T_n$ is the [[Chebyshev polynomial|M6EAR1JHPP]].\nCheck: $2^{1-n} T_n$ is monic, its coefficient of $x^{n-1}$ is $0$ because $T_n$ is even or odd, and no monic polynomial of degree $n$ has a smaller maximum on $[-1, 1]$.\n\nNon-example: for $n = 2$ and $\\sigma = 0$, the polynomial $x^2$ lies in $\\mathcal{Z}_{2,0}$ but is not $Z_{2,0}$.\nIts maximum on $[-1, 1]$ is $1$, while $x^2 - 1/2$ has maximum $1/2$.\n\nSome authors normalise the second coefficient as $-n\\sigma$ instead of $\\sigma$; the family of polynomials is the same.\n",
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