For a finitely generated graded module over a standard graded algebra over a field, with finite-dimensional graded pieces, the Hilbert function is eventually a polynomial in . Its generating series is the Hilbert series. Thus the polynomial describes eventual graded growth, while the series records all graded pieces.
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The concepts of **Hilbert series** and **Hilbert polynomial** arise primarily in algebraic geometry and commutative algebra, particularly in the study of graded algebras and projective varieties. ### Hilbert Series The **Hilbert series** of a graded algebra (or a graded module) is a generating function that encodes the dimensions of its graded components.