Abstract:An algorithm of computing 'b-functions 大阿久俊則 (横浜市大理) IntroductionLet $f(x)\in K[x]=K[x_{1},$ $\ldots,$ $x_{n}1$ be a polynomial with coefficients in a field $I\mathrm{t}'$ of characteristic zero.Let u...An algorithm of computing 'b-functions 大阿久俊則 (横浜市大理) IntroductionLet $f(x)\in K[x]=K[x_{1},$ $\ldots,$ $x_{n}1$ be a polynomial with coefficients in a field $I\mathrm{t}'$ of characteristic zero.Let us denote by $\hat{D}_{n}:=K [[x_{1}, \ldots, x_{n}]]\mathrm{t}\partial_{1},$ $\ldots,$ $\partial_{n})$ the ring of differential operators with formal power series coefficients with $\partial_{i}=\partial/\partial x_{i}$ and $\partial=$ $(\partial_{1}, \ldots, \partial_{n})$ . (If $K$ is a subfield of the field $\mathrm{C}$ of complex numbers, then we can use the ring $D_{\mathfrak{n}}$ of differential operators with convergent power series coefficiets instead of $\hat{D}_{\eta}$ .This makes no difference in the definition below.)Let $s$ be a parameter.Then the (local) $b$ -function (or the Bernstein-Sato polynomial) $b_{f}(s)$ associated with $f(x)$ is the monic polynomial of the least degree $b(s)\in K[s]$ satisfying $P(s, x, \partial)\mathit{4}^{\cdot}(X)s+1=b(s)f(X).$ , with some $P(s, x, \partial)\in D_{n}[s]$ .We present an algorithm of computing the $b$ -function $b_{f}(s)$ for an arbit,rary $f(x)\in I\mathrm{t}'[X]$ .A system $Kan$ of N. Takayama [T2] is available for actual execution of our algolithm.An algorithm of $\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\iota \mathrm{l}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{g}b_{f}(s)$ was first given by M. Sato et al. $1^{\mathrm{s}\mathrm{K}\mathrm{I}\langle 0}$ ] $\backslash \mathrm{v}\mathrm{l}\mathrm{l}\mathrm{e}\mathrm{n}f(?\cdot)\in \mathrm{C}[.\iota]$ is a relative invariant of a prehomogeneous vector space.J. Brian $g\mathrm{o}\mathrm{n}$ et al. [BGMM], [M] have given an algorithm of computing $b_{f}(s)$ for $f(x)\in \mathrm{C}\{x\}$ with isolated singularity.Also note that T. Yano [Y] worked out many interesting examples of b-functions $\mathrm{s}\mathrm{y}_{\mathrm{S}\mathrm{t}\mathrm{e}\ln}\mathrm{a}\mathrm{t}\mathrm{i}_{\mathrm{C}}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{y}$ .Read More