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  "body": "\n## Definition\n\nLet $f \\in \\C[x_1, \\dots, x_n]$ be a nonzero [[polynomial|0SDC934R52]].\nLet $D_n = \\C[x_1, \\dots, x_n]\\langle \\partial_1, \\dots, \\partial_n \\rangle$ be the [[Weyl algebra|98QE7KY55W]] of polynomial differential operators.\nLet $D_n[s]$ be the ring of polynomials in a new central variable $s$ with coefficients in $D_n$.\nThe operators of $D_n[s]$ act on expressions $g \\, f^{s}$ with $g$ in $\\C[x, s, 1/f]$ by the usual rules, with $\\partial_i (f^s) = s \\, (\\partial_i f) \\, f^{s-1}$.\nThe set of polynomials $b(s) \\in \\C[s]$ for which there is $P(s) \\in D_n[s]$ with\n$$P(s) \\, f^{s+1} = b(s) \\, f^s$$\nis an ideal of $\\C[s]$.\nThe *Bernstein–Sato polynomial* $b_f(s)$ is the monic generator of this ideal.\n\n## Remarks\n\nBernstein proved that this ideal is nonzero, so $b_f$ exists.\nFor a nonconstant $f$, $s + 1$ divides $b_f(s)$, as one sees by setting $s = -1$.\n\nExample: $f = x_1^2 + \\cdots + x_n^2$ and $P = \\Delta = \\partial_1^2 + \\cdots + \\partial_n^2$.\nOne has $\\partial_i (f^{s+1}) = 2 (s + 1) x_i f^s$, and then\n$$\\Delta (f^{s+1}) = 2(s+1) \\left( n f^s + 2s \\sum_i x_i^2 f^{s-1} \\right) = 4 (s + 1) \\left( s + \\tfrac{n}{2} \\right) f^s.$$\nSo $(s+1)(s + n/2)$ lies in the ideal; it is in fact $b_f(s)$.\n\nNon-example: for $f = x_1$, the polynomial $(s + 1)(s + 2)$ lies in the ideal, with $P = (s + 2) \\partial_1$, but it is not $b_f$.\nIt is not of least degree, since $\\partial_1 (x_1^{s+1}) = (s+1) x_1^s$ gives $b_f(s) = s + 1$.\n\nThe same definition works for a germ of holomorphic function, with the ring of differential operators with convergent power series coefficients; this gives the local $b$-function at a point.\n",
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