For a single polynomial f (in several variables), we can define the Bernstein-Sato polynomial b(s) as the least-degree, monic polynomial, such that there exists a differential operator P(s) (with polynomial coefficients) that satisfies b(s)f^s=P(s)f^{s+1} [1]. (Such polynomials in fact form a principal ideal, and b is the generator.) It has been described as “a very interesting and delicate invariant of the singularities of {f = 0}” [1].

If we now have p polynomials f₁, ..., f_p, we can generalize Ein et al.’s work [1] to consider polynomials b(s₁, ... ,s_p), such that there exists a differential operator P(s₁, ... ,s_p) (with polynomial coefficients), such that b f₁^s₁ ... f_p^s_]=Pf₁^s₁₊₁... f_p^s_p+1 [2]. Such polynomials still form the (global) Bernstein-Sato ideal, but it is, in general, no longer principal. This ideal is computable [2].

We can also ask about the local Bernstein-Sato ideal at a given point a. This paper gives the first known algorithm for computing the local ideal when p > 1. As an application of this, they explicitly compute the generators in a case where the global ideal is principal, but the local ideal is not. I must confess that it took me two tries to confirm that the local ideal was not principal, bearing out the remark from Ein et al. [1], cited above. The paper presents numerous examples and an interesting heuristic for speeding up the computation.