Bernstein-Sato ideals, associated strati cations, and computer-algebraic aspects

Global and local Bernstein-Sato ideals, Bernstein-Sato polynomials and Bernstein-Sato polynomials of varieties are introduced, their basic properties are proven and their algorithmic determination with the method of Briançon/Maisonobe is presented. Strati cations with respect to the local variants of the introduced polynomials and ideals with the methods of Bahloul/Oaku and Levandovskyy/Martín-Morales are treated and the method of Bahloul/ Oaku is generalized. Moreover, factors of local Bernstein-Sato ideals for disjoint varieties of components, common factors of components and transversally intersecting varieties of components are given. Furthermore, the connection of multivariate and univariate Bernstein-Sato ideals and polynomials B(f1,...,fr) and bf1·...·fr is examined. Budur's approach to determining upper and lower bounds of Bernstein-Sato ideals is presented. Finally, as an application, the computation of annDn(f ) for f ∈ C[x] and α ∈ C is described.