Zolotarev polynomials

Remarks

The minimiser exists and is unique. Indeed $Z_{n,\sigma} = g - q^*$, where $g(x) = x^n + \sigma x^{n-1}$ and $q^*$ is the best approximation of $g$ in the uniform norm on $[-1, 1]$ by polynomials of degree at most $n - 2$.

Example: for $\sigma = 0$ the Zolotarev polynomial is $2^{1-n} T_n$, where $T_n$ is the Chebyshev polynomial. Check: $2^{1-n} T_n$ 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 $[-1, 1]$.

Non-example: for $n = 2$ and $\sigma = 0$, the polynomial $x^2$ lies in $\mathcal{Z}_{2,0}$ but is not $Z_{2,0}$. Its maximum on $[-1, 1]$ is $1$, while $x^2 - 1/2$ has maximum $1/2$.

Some authors normalise the second coefficient as $-n\sigma$ instead of $\sigma$; the family of polynomials is the same.