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  "body": "\n## Summary\n\nThe codimension of a singularity of a smooth function counts how many independent conditions a function must satisfy to have that singularity.\nEquivalently, it is the number of parameters a family of functions needs before the singularity occurs in it in a persistent way.\nPersistent means that small perturbations of the family do not remove it.\nIt is computed from the Jacobian ideal of the [[germ|7SWSRGDT4A]].\nIt is also the least number of parameters of a [[versal unfolding|KSAFWZVB8K]], not counting the constant term.\nIn [[catastrophe theory|6FKJ6SXP3G]] it sorts the elementary catastrophes: the fold has codimension $1$ and the cusp has codimension $2$.\n\n## Explanations\n\n### L1 human\n\nA nondegenerate minimum, like the bottom of the bowl $x^2$, is typical: it survives any small change of the function, so its codimension is $0$.\nA flat inflection like $x^3$ at $0$ is rare: a small change $x^3 + a x$ destroys it unless $a = 0$, so it costs one condition, and its codimension is $1$.\nIt appears unavoidably only in families with at least one parameter.\nThe flatter point $x^4$ needs two parameters, $x^4 + a x^2 + b x$, so its codimension is $2$.\n",
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