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  "body": "\n## Summary\n\nThe Milnor number $\\mu$ of a holomorphic germ $f$ at a critical point is the dimension of its local algebra, see the [[definition|HZCH4KNGY2]].\nThe local algebra is the quotient of the power series ring by the partial derivatives of $f$.\nIt is finite exactly when $f$ has an [[isolated singularity|VTGYEES3VS]].\nIt measures how many simple critical points are merged at the point: a small generic perturbation of $f$ has exactly $\\mu$ nondegenerate critical points nearby.\nTopologically, it is the number of spheres in the [[Milnor fibre|X5HMTTGB62]] of $f$.\n\n## Explanations\n\n### L1 human\n\nA function like $z^2$ has a simple critical point at $0$, while $z^3$ has a worse one.\nIf you perturb $z^3$ to $z^3 - tz$ with small $t \\ne 0$, the critical point splits into two simple ones, at $z = \\pm \\sqrt{t/3}$.\nThe Milnor number counts this: it is $1$ for $z^2$ and $2$ for $z^3$.\nIn general it is computed by pure algebra, as the dimension of a vector space of power series.\n",
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