In order to understand Igusa zeta functions and describe their poles, Maglione—Voll studied a family of functions whose input-polynomials are products of linear forms. In this case, they could express these functions as rational expressions depending only on the combinatorics of the zero set of the product of linear forms. After a natural specialization, the rational expressions simplified even further and Maglione—Voll conjectured the number and multiplicity of the poles of these simplified rational expressions. 

 

In order to prove their conjecture, Maglione, Stump, and I introduced a polynomial simultaneously refining the numerator of Maglione—Voll’s simplified rational expressions, the Poincaré polynomial of a hyperplane arrangement, and the ab-index of a poset. Using this new polynomial and techniques from poset topology, we proved Maglione—Voll’s conjecture and related conjecture from Maglione and Kühne. While their conjecture was stated for geometric lattices, our results turn out to hold for noncrossing partition lattices as well.