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  "body": "\n## Statement\n\nLet $K \\subseteq \\C^n$ be a compact [[polynomially convex set|AD3474PVHR]].\nLet $f$ be a [[holomorphic function|507C34W0ZT]] on an open set containing $K$.\nThen for every $\\varepsilon \u003e 0$ there is a [[polynomial|07GNX8GCYP]] $p$ in $z_1, \\dots, z_n$ with $\\abs{f(z) - p(z)} \u003c \\varepsilon$ for all $z \\in K$.\nThat is, $f$ is a [[uniform limit|08VZ57PT26]] of polynomials on $K$.\n\n## Remarks\n\nThe proof first finds polynomials $p_1, \\dots, p_N$ such that the polynomial polyhedron $P = \\set{z : \\abs{z_j} \\le R, \\ \\abs{p_k(z)} \\le 1}$ contains $K$ and lies in the open set where $f$ is defined; this uses polynomial convexity of $K$.\nThe map $z \\mapsto (z, p_1(z), \\dots, p_N(z))$ embeds a neighbourhood of $P$ as a closed complex submanifold of a polydisc in $\\C^{n+N}$.\nOka's extension theorem extends $f$ from this submanifold to a holomorphic function on the polydisc, and the Taylor series of the extension gives polynomials converging to $f$ uniformly on $K$.\n\nExample: for $n = 1$ the theorem is the polynomial case of [[Runge's theorem|6DG4CTGXJD]]: if $\\C \\setminus K$ is connected, every function holomorphic near $K$ is a uniform limit of polynomials on $K$.\n\nPolynomial convexity cannot be dropped.\nOn the unit circle $K$, which is not polynomially convex, the function $1/z$ is not a uniform limit of polynomials: polynomials $p$ satisfy $\\int_K p(z) \\, dz = 0$, while $\\int_K z^{-1} \\, dz = 2\\pi i$.\n\n## Explanations\n\n### L1 human\n\nOn a compact set that equals its own polynomial hull, every holomorphic function can be approximated as closely as one likes by polynomials.\nThis is the several-variable version of Runge's approximation theorem.\nThe condition on the set rules out holes that would trap functions such as $1/z$.\n\n### L2 human\n\nThe theorem is a basic approximation result of several complex variables.\nIt generalises to holomorphically convex compact subsets of Stein manifolds and of domains of holomorphy, with polynomials replaced by holomorphic functions on the whole domain.\nThe method of its proof, embedding a polynomial polyhedron into a polydisc, reduces global questions to questions on polydiscs.\nIt is also used to show that convex domains and balls are Runge domains.\n",
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