Bernstein–Sato polynomial

Summary

The Bernstein–Sato polynomial, or $b$-function, of a polynomial $f$ is a monic polynomial $b(s)$, see the definition. It is the one of least degree for which a differential operator turns $f^{s+1}$ into $b(s) f^s$. Joseph Bernstein proved its existence for polynomials. Mikio Sato introduced it independently in his theory of prehomogeneous vector spaces. Masaki Kashiwara proved that its roots are negative rational numbers. It is a fine invariant of the singularities of $f^{-1}(0)$. Its roots are linked with the eigenvalues of the monodromy and with the log canonical threshold. It also gives the meromorphic continuation of the distribution $\abs{f}^s$ in $s$.