In mathematics, and in particular in the field of algebra, a Hilbert–Poincaré series (also known under the name Hilbert series), named after David Hilbert and Henri Poincaré, is an adaptation of the notion of dimension to the context of graded algebraic structures (where the dimension of the entire structure is often infinite). It is a formal power series in one indeterminate, say t {\displaystyle t} , where the coefficient of t n {\displaystyle t^{n}} gives the dimension (or rank) of the sub-structure of elements homogeneous of degree n {\displaystyle n} . That is, it is a Generating function of the dimensions. It is closely related to the Hilbert polynomial in cases when the latter exists; however, the Hilbert–Poincaré series describes the rank in every degree, while the Hilbert polynomial describes it only in all but finitely many degrees, and therefore provides less information. In particular the Hilbert–Poincaré series cannot be deduced from the Hilbert polynomial even if the latter exists. In good cases, the Hilbert–Poincaré series can be expressed as a rational function of its argument t {\displaystyle t} .
Definition Let K be a field, and let V = ⨁ i ∈ N V i {\displaystyle V=\textstyle \bigoplus _{i\in \mathbb {N} }V_{i}} be an N {\displaystyle \mathbb {N} } -graded vector space over K, where each subspace V i {\displaystyle V_{i}} of vectors of degree i is finite-dimensional. Then the Hilbert–Poincaré series of V is the formal power series
∑ i ∈ N dim K ( V i ) t i . {\displaystyle \sum _{i\in \mathbb {N} }\dim _{K}(V_{i})t^{i}.}
A similar definition can be given for an N {\displaystyle \mathbb {N} } -graded R-module over any commutative ring R in which each submodule of elements homogeneous of a fixed degree n is free of finite rank; it suffices to replace the dimension by the rank. Often the graded vector space or module of which the Hilbert–Poincaré series is considered has additional structure, for instance, that of a ring, but the Hilbert–Poincaré series is independent of the multiplicative or other structure. Example: Since there are ( n + k k ) {\displaystyle \textstyle {\binom {n+k}{k}}} monomials of degree k in variables X 0 , … , X n {\displaystyle X_{0},\dots ,X_{n}} (by induction, say), one can deduce that the sum of the Hilbert–Poincaré series of K [ X 0 , … , X n ] {\displaystyle K[X_{0},\dots ,X_{n}]} is the rational function 1 / ( 1 − t ) n + 1 {\displaystyle 1/(1-t)^{n+1}} .
Hilbert–Serre theorem Suppose M is a finitely generated graded module over A [ x 1 , … , x n ] , deg x i = d i {\displaystyle A[x_{1},\dots ,x_{n}],\deg x_{i}=d_{i}} with an Artinian ring (e.g., a field) A. Then the Poincaré series of M is a polynomial with integral coefficients divided by ∏ ( 1 − t d i ) {\displaystyle \prod (1-t^{d_{i}})} . The standard proof today is an induction on n. Hilbert's original proof made a use of Hilbert's syzygy theorem (a projective resolution of M), which gives more homological information. Here is a proof by induction on the number n of indeterminates. If n = 0 {\displaystyle n=0} , then, since M has finite length, M k = 0 {\displaystyle M_{k}=0} if k is large enough. Next, suppose the theorem is true for n − 1 {\displaystyle n-1} and consider the exact sequence of graded modules (exact degree-wise), with the notation N ( l ) k = N k + l {\displaystyle N(l)_{k}=N_{k+l}} ,
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