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  "body": "\n## Definition\n\nLet $K \\subseteq \\C^n$ be a [[compact set|0R6EWFKRCF]].\nIts *polynomially convex hull* is\n$$\\hat K = \\set{z \\in \\C^n : \\abs{p(z)} \\le \\max_{w \\in K} \\abs{p(w)} \\text{ for every polynomial } p \\in \\C[z_1, \\dots, z_n]},$$\nwhere the polynomials are the [[polynomials in several variables|07GNX8GCYP]] with complex coefficients.\nThe set $K$ is *polynomially convex* if $\\hat K = K$.\n\n## Remarks\n\nAlways $K \\subseteq \\hat K$, and $\\hat K$ is compact.\nIt coincides with the [[holomorphically convex hull|CD6SMMZSW8]] of $K$ in $D = \\C^n$, because entire functions are uniform limits of their Taylor polynomials on compact sets.\n\nExample: a compact convex set $K$ is polynomially convex.\nIf $z \\notin K$, there is a complex linear function $\\ell$ with $\\operatorname{Re} \\ell(z) \u003e \\max_K \\operatorname{Re} \\ell$.\nThen $f = e^{\\ell}$ satisfies $\\abs{f(z)} \u003e \\max_K \\abs{f}$, and the Taylor polynomials of $f$ converge to $f$ uniformly on $K \\cup \\set{z}$, so some polynomial $p$ has $\\abs{p(z)} \u003e \\max_K \\abs{p}$.\n\nNon-example: the unit circle $K = \\set{\\abs{z} = 1}$ in $\\C$.\nBy the maximum principle, $\\abs{p(0)} \\le \\max_K \\abs{p}$ for every polynomial $p$, so $0 \\in \\hat K$ but $0 \\notin K$.\n\nFor $n = 1$, a compact set $K \\subseteq \\C$ is polynomially convex exactly when $\\C \\setminus K$ is connected.\n",
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