Validity of Budur’s conjecture for principal monoid ideals

Determine whether Budur’s original conjecture that Bernstein–Sato ideals along monoid ideals are generated by products of linear polynomials remains valid for principal monoid ideals; the exhibited counterexample uses the nonprincipal monoid ideal K={3e_1,3e_2} and therefore does not resolve the principal case.

Background

The paper studies Bernstein–Sato ideals B^K_f associated with an r-tuple of functions and a monoid ideal K in N^r. Budur’s conjecture, as summarized in the introduction, predicts that such ideals are generated by products of linear polynomials of the form c_1s_1+⋯+c_rs_r+c_0, with nonnegative integer coefficients c_i and positive constant term c_0.

The paper constructs a reduced free arrangement in C^2, namely f=(x,y,x+y,x+2y), and computes the Bernstein–Sato ideal for the nonprincipal monoid ideal K={3e_1,3e_2}. Its zero locus contains an irreducible nonlinear quadric, disproving the conjecture for this nonprincipal K. The authors explicitly note that this does not settle whether the original conjecture holds in the principal case.