We prove the rationality of the Poincar\'e series of multiplier ideals in any dimension and thus extending the main results for surfaces of Galindo and Monserrat and Alberich-Carrami\~nana et al. Our results also hold for Poincar\'e series of test ideals. In order to do so, we introduce a theory of Hilbert functions indexed over $\mathbb{ R}$ which gives an unified treatment of both cases.
Categories: math.AC math.AG