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  "body": "\n## Summary\n\nThe ring of convergent power series $\\C\\{z_1, \\dots, z_n\\}$ consists of the formal power series in $n$ variables that converge on some neighbourhood of the origin.\nIt is a subring of the ring of [[multivariate formal power series|3E8ZG3NJYZ]] $\\C[\\![z_1, \\dots, z_n]\\!]$.\nIt is isomorphic to the ring of [[germs of holomorphic functions|7PJ8KVZZCQ]] at $0 \\in \\C^n$, and it is the basic algebraic object of local analytic geometry.\nIt is a Noetherian local unique factorization domain, as is its formal completion.\n\n## Explanations\n\n### L1 human\n\nA power series in several variables is an infinite sum of monomials $z_1^{k_1} \\cdots z_n^{k_n}$ with coefficients.\nSome such series converge near the origin, such as $\\sum_{k} (z_1 z_2)^k$, and some converge only at the origin, such as $\\sum_k k!\\, z_1^k$.\nThe convergent ones can be added and multiplied, and they form a ring.\nThis ring encodes all holomorphic functions near the origin, through their Taylor series.\n",
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