Geometry of Bernstein-Sato ideals

In studying Mellin transforms of multivariable polynomial functions, Gelfand defined the so-called Archimedean zeta function of a polynomial and conjectured that the archimedean zeta function has a meromorphic continuation on the whole complex plane in 1950s. Bernstein introduced the so-called Bernstein-Sato polynomial (or b-function) and solved the conjecture in the 1970s. In this talk, I will discuss how we can generalize the construction of Bernstein for a finite union of polynomial functions by defining Bernstein-Sato ideals following the ideas of Sabbah. Then I will discuss geometric properties of such ideals and prove that the variety of the Bernstein-Sato ideal is defined over Q and each of its irreducible components is a translated linear subspace, generalizing a classical result of Kashiwara for b-functions.