Hilbert–Poincaré series

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In mathematics, and in particular in the field of algebra, a Hilbert–Poincaré 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, where the coefficient of tn gives the dimension (or rank) of the sub-structure of elements homogeneous of degree n.

[edit] Definition

Let K be a field, and let V=\textstyle\bigoplus_{i\in\mathbf{N}}V_i be a N-graded vector space over K, where each subspace Vi of vectors of degree n is finite dimensional. Then the Hilbert–Poincaré series of V is the formal power series

\sum_{i\in\mathbf{N}}\dim_K(V_i)t^i.

A similar definition can be given for an 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.

[edit] Examples

A basic example of a graded vector space is the polynomial ring K[X], graded by degree. Since the monomials (Xi)i∈N form a basis of the underlying vector space, formed of homogeneous elements, the dimension of each homogeneous component is 1, and the Hilbert–Poincaré series of K[X] is therefore

\sum_{i\in\mathbf{N}}t^i = \frac1{1-t}.

For a polynomial ring K[X,Y] in two indeterminates, graded by total degree, the monomials again form a vector space basis formed of homogeneous elements; this time there are d + 1 monomials of degree d for any d, namely Xd, Xd–1Y, Xd–2Y2, …, Yd. Therefore the Hilbert–Poincaré series of K[X,Y] is

\sum_{i\in\mathbf{N}}(i+1)t^i = \frac1{(1-t)^2}.

The fact that this is the square of the Hilbert–Poincaré series of K[X] is no accident: one has K[X,Y] = K[X] ⊗ K[Y] as graded rings, and in general the Hilbert–Poincaré series of the tensor product of graded vector spaces is the product of their the Hilbert–Poincaré series. It follows that the the Hilbert–Poincaré series of K[X1,X2,…,Xn] is

\frac1{(1-t)^n}

for any n.