Motivic zeta functions in additive monoidal categories

Abstract

Let C be a pseudo-abelian symmetric monoidal category, and X a Schur-finite object of C . We study the problem of rationality of the motivic zeta function ζ x(t) of X . Since the coefficient ring is not a field, there are several variants of rationality — uniform, global, determinantal and pointwise rationality. We show that ζ x(t) is determinantally rational, and we give an example of C and X for which the motivic zeta function is not uniformly rational.