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It is April 14 and the Sony PSN or PlayStation Network down status mills about. And while the service struggles to stand up and again online, a non-public letter to PS3 publishers leaked into the public domain.<br><br>
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 ''t''<sup>''n''</sup> gives the dimension (or rank) of the sub-structure of elements homogeneous of degree&nbsp;''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 in 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&nbsp;''t''.


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Let ''K'' be a field, and let <math>V=\textstyle\bigoplus_{i\in\mathbf{N}}V_i</math> be a '''N'''-[[graded vector space]] over ''K'', where each subspace ''V''<sub>''i''</sub> of vectors of degree ''n'' is finite dimensional. Then the Hilbert–Poincaré series of ''V'' is the [[formal power series]]
:<math>\sum_{i\in\mathbf{N}}\dim_K(V_i)t^i.</math>
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 module|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 <math>\binom {n+k}{n}</math> monomials of degree ''k'' in variables <math>X_0, \dots, X_n</math> (by induction, say), it follows immediately that the Hilbert–Poincaré series of ''K''[''X''<sub>0</sub>,''X''<sub>1</sub>,…,''X''<sub>''n''</sub>] is <math>(1-t)^{-n-1}</math>
 
== Hilbert-Serre theorem ==
Suppose ''M'' is a finitely generated graded module over <math>A_0[x_0, \dots, x_n], \operatorname{deg}x_i = d_i, A_0</math> artinian. Then the Poincaré series of ''M'' is a polynomial divided by <math>\prod (1-t^{d_i})</math>.
 
The proof is by induction on ''n''. If <math>n =0</math>, then <math>M_k = 0</math> if ''k'' is large enough. Next, suppose the theorem is true for <math>n - 1</math> and consider the exact sequence, with the notation <math>N(l)_k = N_{k+l}</math>,
:<math>0 \to K \to M \overset{x_n} \to M(d_n) \to C(d_n) \to 0</math>.
Since the length is additive, Poincaré series are also additive. Hence, we have:
:<math>P(M, t) = P(K, t) + P(M(d_n), t) - P(C(d_n), t)</math>.
We can write <math>P(M(d_n), t) = t^{d_n} P(M, t) + </math> a polynomial. Since ''K'' is killed by <math>x_n</math>, we can regard it as a graded module over <math>A_0[x_0, \dots, x_{n-1}]</math>; the same for ''C''. The theorem thus now follows from the inductive hypothesis.
 
== Chain complex ==
 
An example of graded vector space is associated to a [[chain complex]], or cochain complex ''C'' of vector spaces; the latter takes the form
 
:<math> 0\to C^0 \stackrel{d_0}{\longrightarrow} C^1\stackrel{d_1}{\longrightarrow} C^2 \stackrel{d_2}{\longrightarrow} \cdots \stackrel{d_{n-1}}{\longrightarrow} C^n \longrightarrow 0. </math>
 
The Hilbert–Poincaré series (here often called the Poincaré polynomial) of the graded vector space <math>\bigoplus_iC^i</math> for this complex is
 
:<math>P_C(t) = \sum_{j=0}^n \dim(C^j)t^j.</math>
 
The Hilbert-Poincaré polynomial of the [[cohomology]], with cohomology spaces ''H''<sup>''j''</sup>&nbsp;=&nbsp;''H''<sup>''j''</sup>(''C''), is
 
:<math>P_H(t) = \sum_{j=0}^n \dim(H^j)t^j.</math>
 
A famous relation between the two is that there is a polynomial <math>Q(t)</math> with non-negative coefficients, such that <math>P_C(t) - P_H(t) = (1+t)Q(t).</math>
 
{{DEFAULTSORT:Hilbert-Poincare series}}
[[Category:Abstract algebra]]
[[Category:Linear algebra]]
[[Category:Mathematical series]]

Latest revision as of 18:51, 20 November 2014

It is April 14 and the Sony PSN or PlayStation Network down status mills about. And while the service struggles to stand up and again online, a non-public letter to PS3 publishers leaked into the public domain.

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