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  "body": "\n## Summary\n\nThe Bernstein–Sato polynomial, or $b$-function, of a polynomial $f$ is a monic polynomial $b(s)$, see the [[definition|R2AV47AX0Y]].\nIt is the one of least degree for which a differential operator turns $f^{s+1}$ into $b(s) f^s$.\nJoseph Bernstein proved its existence for polynomials.\nMikio Sato introduced it independently in his theory of prehomogeneous vector spaces.\nMasaki Kashiwara proved that its roots are negative rational numbers.\nIt is a fine invariant of the singularities of $f^{-1}(0)$.\nIts roots are linked with the eigenvalues of the [[monodromy|DQP63REKEK]] and with the log canonical threshold.\nIt also gives the meromorphic continuation of the distribution $\\abs{f}^s$ in $s$.\n\n## Explanations\n\n### L1 human\n\nFor $f(x) = x$, differentiating $x^{s+1}$ gives $(s+1) x^s$.\nSo one derivative turns $x^{s+1}$ into $x^s$ times the polynomial $s + 1$.\nBernstein showed that for every polynomial $f$ in any number of variables there is a differential operator doing the same job, up to a factor $b(s)$.\nThe best possible factor is the Bernstein–Sato polynomial, and its roots record how singular $f$ is.\n",
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