Hilbert series and Hilbert polynomial
= Hilbert series and Hilbert polynomial
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For a finitely generated graded module over a standard graded algebra over a <field>, with finite-dimensional graded pieces, the <Hilbert function> $h(n)=\dim M_n$ is eventually a polynomial in $n$. Its generating series is the <Hilbert series>. Thus the polynomial describes eventual graded growth, while the series records all graded pieces.