半擬斉次孤立特異点の性質を利用した局所$b$-関数の計算

Abstract

Local b-functions (Bernstein-Sato polynomials) are one of important invariants of hypersurface singularities. In this paper, semi-quasihomogeneous hypersurface singularities and their local bfunctions are considered. New computation methods of Ann(fs) and local b-functions of the semiquasihomogeneous hypersurface singularities are introduced. The underlining ideas of the new methods are using properties of semi-quasihomogeneous hypersurface singularities and generalized integral dependence relations. It is shown that Ann(fs) and local b-functions can be obtained in a commutative polynomial ring, namely, non-commutative Gröbner bases are not needed in the new method. The resulting algorithms of the proposed new methods have been implemented in the compute algebra system Risa/ Asir. The results of benchmark tests are also given.