Brian\c{c}on-Skoda exponents and the maximal root of reduced Bernstein-Sato polynomials

For a holomorphic function $f$ on a complex manifold $X$, the Brian\c{c}on-Skoda exponent $e^{\rm BS}(f)$ is the smallest integer $k$ with $f^k\in(\partial f)$ (replacing $X$ with a neighborhood of $f^{-1}(0)$), where $(\partial f)$ denotes the Jacobian ideal of $f$. It is shown that $e^{\rm BS}(f)\le d_X$ $(:=\dim X)$ by Brian\c con-Skoda. We prove that $e^{\rm BS}(f)\le[d_X-2\widetilde{\alpha}_f]+1$ with $-\widetilde{\alpha}_f$ the maximal root of the reduced Bernstein-Sato polynomial $b_f(s)/(s+1)$, assuming the latter exists (shrinking $X$ if necessary). This implies for instance that $e^{\rm BS}(f)\le d_X-2$ in the case $f^{-1}(0)$ has only rational singularities, that is, if $\widetilde{\alpha}_f>1$.