Which series are Hilbert series of graded modules over polynomial rings?
  Let $S$ be a multigraded polynomial ring such that the degree of each variable is a unit vector; so $S$ is the homogeneous coordinate ring of a product of projective spaces. In this setting, we characterize the formal Laurent series which arise as Hilbert series of finitely generated $S$-modules.   Also we provide necessary conditions for a formal Laurent series to be the Hilbert series of a finitely generated module with a given depth. In the bigraded case (corresponding to the product of two projective spaces), we completely classify the Hilbert series of finitely generated modules of positive depth. 
Lukas Katth\"an, Julio Jos\'e Moyano-Fern\'andez, Jan Uliczka
math.AC math.CO