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  "body": "\n## Summary\n\nThe Siciak extremal function $V_E$ of $E \\subseteq \\C^n$ is the supremum of all functions of the [[Lelong class|X4ZHQG9995]] that are at most $0$ on $E$.\nFor compact $E$ it can also be computed from polynomials.\nIt is the supremum of $\\frac{1}{\\deg p} \\log\\abs{p}$ over polynomials $p$ of degree at least $1$ with $\\abs{p} \\le 1$ on $E$.\nIt is the higher-dimensional version of the Green function of a planar compact set with pole at infinity.\nIt is also called the pluricomplex Green function of $E$ with pole at infinity.\nIt controls polynomial approximation and interpolation on $E$, and for compact $E$ its upper semicontinuous regularisation is identically $+\\infty$ exactly when $E$ is [[pluripolar|N0MAGPW8AP]].\n\n## Explanations\n\n### L1 human\n\nSuppose a polynomial is at most $1$ in size on a set $E$.\nHow large can it be at a point $z$ outside $E$, given its degree?\nThe Siciak extremal function answers this: the best bound is $\\abs{p(z)} \\le e^{d \\, V_E(z)}$ for polynomials of degree $d$.\nFor the closed unit ball in $\\C^n$, the answer is $V_E(z) = \\log\\abs{z}$ outside the ball.\nSo a polynomial of degree $d$ bounded by $1$ on the ball is bounded by $\\abs{z}^d$ outside it.\n",
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