  Absolutely Koszul algebras are a class of rings over which any finite graded
module has a rational Poincar\'e series. We provide a criterion to detect
non-absolutely Koszul rings. Combining the criterion with machine computations,
we identify large families of Veronese subrings and Segre products of
polynomial rings which are not absolutely Koszul. In particular, we classify
completely the absolutely Koszul algebras among Segre products of polynomial
rings, at least in characteristic $0$.
