Hilbert–Poincaré series

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, where the coefficient of tⁿ gives the dimension (or rank) of the sub-structure of elements homogeneous of degree n. 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.

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

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