`oaku@lab.twcu.ac.jp`

For a real polnomial $f$ in several variables, one can define the complex power $f_+^s$ as a distribution (a generalized function) with a holomorphic parameter $s$. Bernstein proved that this distribution can be extended as a distribution-valued meromorphic function of $s$ on the whole complex plane by using what is called the Bernstein-Sato polynomial or the $b$-function, which plays an essential role in $D$-module theory as well as in singularity theory.

Hence it is an interesting problem to determine the poles of $f_+^s$ and the Laurent expansion around each pole. I will present some examples of, as well as algorithms to compute, holonomic systems for Laurent coefficients of $f_+^s$. I will also mention some relation with the local cohomology group associated with $f$.