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| Given a [[graded algebra|graded commutative algebra]] finitely generated over a [[field (mathematics)|field]], the '''[[David Hilbert|Hilbert]] function''', '''Hilbert polynomial''', and '''Hilbert series''' are three strongly related notions which measure the growth of the dimension of its homogeneous components.
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| These notions have been extended to [[filtered algebra]]s and graded filtered [[modules]] over these algebras.
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| The typical situations where these notions are used are the following:
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| * The quotient by a homogeneous [[Ideal (ring theory)|ideal]] of a [[multivariate polynomial]] [[polynomial ring|ring]], graded by the total degree.
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| * The quotient by an [[Ideal (ring theory)|ideal]] of a [[multivariate polynomial]] [[polynomial ring|ring]], filtered by the total degree.
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| * The filtration of a [[local ring]] by the powers of its [[maximal ideal]]. In this case the Hilbert polynomial is called the [[Hilbert–Samuel function|Hilbert–Samuel polynomial]].
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| The Hilbert series of an algebra or a module is a special case of the [[Hilbert–Poincaré series]] of a [[graded vector space]].
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| Hilbert polynomial and Hilbert series are important in computational [[algebraic geometry]], as they are the easiest known way for computing the dimension and the degree of an algebraic variety defined by explicit polynomial equations.
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| == Definitions and main properties==
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| Let us consider a finitely generated [[graded algebra|graded commutative algebra]] ''S'' over a [[field (mathematics)|field]] ''K'', which is finitely generated by elements of positive degree. This means that
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| :<math>S = \bigoplus_{i \ge 0} S_i\ </math>
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| and that <math>S_0=K</math>.
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| The '''Hilbert function'''
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| :<math>HF_S\;:\;n\mapsto \dim_K\,S_n</math>
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| maps the integer ''n'' onto the dimension of the ''K''-vector space ''S''<sub>''n''</sub>. The '''Hilbert series''', which is called [[Hilbert–Poincaré series]] in the more general setting of graded vector spaces, is the [[formal series]]
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| :<math>HS_S(t)=\sum_{n=0}^{\infty} HF_S(n)\,t^n.</math>
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| If ''S'' is generated by ''h'' homogeneous elements of positive degrees <math>d_1, \ldots, d_h</math>, then the sum of the Hilbert series is a rational fraction
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| :<math>HS_S(t)=\frac{Q(t)}{\prod_{i=1}^h (1-t^{d_i})}\,,</math>
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| where ''Q'' is a polynomial with integer coefficients.
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| If ''S'' is generated by elements of degrees 1 then the sum of the Hilbert series may be rewritten as
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| :<math>HS_S(t)=\frac{P(t)}{(1-t)^\delta}\,,</math>
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| where ''P'' is a polynomial with positive integer coefficients.
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| In this case the series expansion of this rational fraction is
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| :<math>HS_S(t)=P(t)\,\left(1+\delta\,t+\cdots +\binom{n+\delta-1}{\delta-1}\,t^n+\cdots\right)</math>
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| where the [[binomial coefficient]] <math>\binom{n+\delta-1}{\delta-1}</math> is <math>\;\frac{(n+\delta-1)(n+\delta-2)\cdots n}{(\delta-1)!}\;</math> for <math>n>-\delta</math> and 0 otherwise.
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| This shows that there exists a unique polynomial <math>HP_S(n)</math> with rational coefficients which is equal to <math>HF_S(n)</math> for <math> n\ge \deg P-\delta+1</math>. This polynomial is the '''Hilbert polynomial'''. The least ''n''<sub>0</sub> such that <math>HP_S(n)=HF_S(n)</math> for ''n'' ≥ ''n''<sub>0</sub> is called the '''Hilbert regularity'''. It may be lower than <math>\deg P-\delta+1</math>.
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| The Hilbert polynomial is a [[numerical polynomial]], since the dimensions are integers, but the polynomial almost never has integer coefficients {{harv|Schenck|2003|pp=41}}.
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| All these definitions may be extended to finitely generated [[graded module]]s over ''S'', with the only difference that a factor ''t''<sup>''m''</sup> appears in the Hilbert series, where ''m'' is the minimal degree of the generators of the module, which may be negative.
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| The '''Hilbert function''', the '''Hilbert series''' and the '''Hilbert polynomial''' of a [[filtered algebra]] are those of the associated graded algebra.
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| The Hilbert polynomial of a [[projective variety]] ''V'' in '''P'''<sup>''n''</sub> is defined as the Hilbert polynomial of the [[homogeneous coordinate ring]] of ''V''.
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| == Graded algebra and polynomial rings ==
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| Polynomial rings and their quotients by homogeneous ideals are typical graded algebras. Conversely, if ''S'' is a graded algebra generated over the field ''K'' by ''n'' homogeneous elements ''g''<sub>1</sub>, ..., ''g''<sub>''n''</sub> of degree 1, then the map which sends ''X''<sub>''i''</sub> onto ''g''<sub>''i''</sub> defines an homomorphism of graded rings from <math>R_n=K[X_1,\ldots, X_n]</math> onto ''S''. Its [[Kernel (algebra)|kernel]] is a homogeneous ideal ''I'' and this defines an isomorphism of graded algebra between <math>R_n/I</math> and ''S''.
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| Thus, the graded algebras generated by elements of degree 1 are exactly, up to an isomorphism, the quotients of polynomial rings by homogeneous ideals. Therefore, the remainder of this article will be restricted to the quotients of polynomial rings by ideals.
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| == Properties of Hilbert series ==
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| === Additivity ===
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| Hilbert series and Hilbert polynomial are additive relatively to [[exact sequence]]s. More precisely, if
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| :<math>0 \;\rightarrow\; A\;\rightarrow\; B\;\rightarrow\; C \;\rightarrow\; 0</math> | |
| is an exact sequence of graded or filtered modules, then we have
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| :<math>HS_B=HS_A+HS_C</math>
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| and
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| :<math>HP_B=HP_A+HP_C.</math>
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| This follows immediately from the same property for the dimension of vector spaces.
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| === Quotient by a non-zero divisor ===
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| Let ''A'' be a graded algebra and ''f'' a homogeneous element of degree ''d'' in ''A'' which is not a [[zero divisor]]. Then we have
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| :<math>HS_{A/f}(t)=(1-t^d)\,HS_A(t)\,.</math>
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| It follows immediately from the additivity on the exact sequence
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| :<math>0 \;\rightarrow\; A^{[d]}\; \xrightarrow{f}\; A \;\rightarrow\; A/f\rightarrow\; 0\,,</math>
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| where the arrow labeled ''f'' is the multiplication by ''f'' and <math>A^{[d]}</math> is the graded algebra, which is obtained from ''A'' by shifting the degrees by ''d'', in order that the multiplication by ''f'' has degree 0. This implies that <math>HS_{A^{[d]}}(t)=t^d\,HS_A(t)\,.</math>
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| === Hilbert series and Hilbert polynomial of a polynomial ring === | |
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| The Hilbert series of the polynomial ring <math>R_n=K[x_0, \ldots, x_n]</math> is
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| :<math>HS_{R_n}(t) = \frac{1}{(1-t)^{n+1}}\,.</math>
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| It follows that the Hilbert polynomial is
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| : <math> HP_{R_n}(k) = {{k+n}\choose{n}} = \frac{(k+1)\cdots(k+n)}{n!}\,.</math>
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| The proof that the Hilbert series has this simple form is obtained by applying recursively the previous formula for the quotient by a non zero divisor (here <math>x_n</math>) and remarking that <math>HS_K(t)=1\,.</math>
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| === Shape of the Hilbert series and dimension ===
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| A graded algebra ''A'' generated by homogeneous elements of degree 1 has [[Krull dimension]] zero if the maximal homogeneous ideal, that is the ideal generated by the homogeneous elements of degree 1, is [[nilpotent ideal|nilpotent]]. This implies that the dimension of ''A'' as a ''K''-vector space is finite and the Hilbert series of ''A'' is a polynomial ''P''(''t'') such that ''P''(1) is equal to the dimension of ''A'' as a ''K'' vector space.
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| If the Krull dimension of ''A'' is positive, there is a homogeneous element ''f'' of degree one which is not a zero divisor (in fact almost all elements of degree one have this property). The Krull dimension of ''A''/''f'' is the Krull dimension of ''A'' minus one.
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| The additivity of Hilbert series shows that <math>HS_{A/f}(t)=(1-t)\,HS_A(t)</math>. Iterating this a number of times equal to the Krull dimension of ''A'', we get eventually an algebra of dimension 0 whose Hilbert series is a polynomial ''P''(''t''). This show that the Hilbert series of ''A'' is
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| :<math>HS_A(t)=\frac{P(t)}{(1-t)^d}</math>
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| where the polynomial ''P''(''t'') is such that ''P''(1) ≠ 0 and ''d'' is the Krull dimension of ''A''.
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| This formula for the Hilbert series implies that the degree of the Hilbert polynomial is ''d'' and that its leading coefficient is ''P''(1)/''d''!.
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| == Degree of a projective variety and Bézout's theorem ==
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| The Hilbert series allows us to compute the [[degree of an algebraic variety]] as the value at 1 of the numerator of the Hilbert series. This provides also a simple proof of [[Bézout's theorem]]. For this purpose, let us consider an [[projective algebraic set]] {{mvar|''V''}} defined as the set of the zeros of a [[homogeneous ideal]] <math>I\subset k[x_0, x_1, \ldots, x_n]</math>, where {{mvar|''k''}} is a field, and let <math> R=k[x_1, \ldots, x_n]/I</math> be the ring of the [[regular function]]s on the algebraic set (in this section, we do not need that the algebraic set be irreducible nor that the ideal is prime).
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| If the dimension of {{mvar|''V''}}, equal to the dimension of {{mvar|''R''}} is {{mvar|d}}, the degree of {{mvar|''V''}} is the number of points of intersection, counted with multiplicity, of {{mvar|''V''}} with the intersection of <math>d</math> hyperplanes in [[general position]]. This implies that the equations of these hyperplanes, say <math>h_1, \ldots, h_{d},</math> are a [[regular sequence]], and that we have the exact sequences
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| :<math>0 \;\rightarrow\; \left(R/\langle h_1,\ldots, h_{k-1}\rangle \right)^{[1]}\; \xrightarrow{h_k}\; R/\langle h_1,\ldots, h_{k-1}\rangle \;\rightarrow\; R/\langle h_1,\ldots, h_{k}\rangle\;\rightarrow\; 0,</math>
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| for <math>k=1, \ldots, d.</math> This implies that
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| :<math>HS_{R/\langle h_1,\ldots, h_{d}\rangle}(t) = (1-t)^d\,HS_R(t)</math>
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| is a polynomial, which is equal to the numerator <math> P(t)</math> of the Hilbert series of {{mvar|R}}. After dehomogeneizing by putting <math>x_0=1</math>, [[Jordan-Hölder theorem]] for [[Artinian ring]]s allows to prove that <math>P(1)</math> is the degree of the algebraic set {{mvar|''V''}}.
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| Similarly, if {{mvar|''f''}} is a homogeneous polynomial of degree <math>\delta</math>, which is not a zero divisor in {{mvar|''R''}}, the exact sequence
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| :<math>0 \;\rightarrow\; R^{[\delta]}\; \xrightarrow{f}\; R \;\rightarrow\; R/\langle f\rangle\;\rightarrow\; 0,</math>
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| shows that
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| :<math>HS_{R/\langle f \rangle}(t)=(1-t^\delta)HS_R(t).</math>
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| Looking on the numerators this proves the following generalization of Bézout's theorem.theorem:
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| ''If'' {{mvar|''f''}} ''is a homogeneous polynomial of degree'' <math>\delta</math>, ''which is not a zero divisor in'' {{mvar|''R''}}, ''then the degree of the intersection of'' {{mvar|''V''}} ''with the hypersurface defined by'' {{mvar|''f''}} ''is the product of the degree of'' {{mvar|''V''}} ''by'' <math>\delta</math> ''.
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| The usual Bézout's theorem is easily deduced by starting from a hypersurface and intersecting it, one after the other, with <math>n-1</math> other hypersurfaces.
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| == Computation of Hilbert series and Hilbert polynomial ==
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| The Hilbert polynomial is easily deducible from the Hilbert series. This section describes how the Hilbert series may be computed in the case of a quotient of a polynomial ring, filtered or graded by the total degree.
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| Thus let ''K'' a field, <math>R=K[x_1,\ldots,x_n]</math> be a polynomial ring and ''I'' be an ideal in ''R''. Let ''H'' be the homogeneous ideal generated by the homogeneous parts of highest degree of the elements of ''I''. If ''I'' is homogeneous, then ''H''=''I''. Finally let ''B'' be a [[Gröbner basis]] of ''I'' for a [[monomial ordering]] refining the [[total degree]] partial ordering and ''G'' the (homogeneous) ideal generated by the leading monomials of the elements of ''B''.
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| The computation of the Hilbert series is based on the fact that ''the filtered algebra R/I and the graded algebras R/H and R/G have the same Hilbert series''.
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| Thus the computation of the Hilbert series is reduced, through the computation of a Gröbner basis, to the same problem for an ideal generated by monomials, which is usually much easier than the computation of the Gröbner basis. The [[computational complexity]] of the whole computation depends mainly on the regularity, which is the degree of the numerator of the Hilbert series. In fact the Gröbner basis may be computed by linear algebra over the polynomials of degree bounded by the regularity.
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| The computation of Hilbert series and Hilbert polynomials are available in most [[computer algebra system]]s. For example in both [[Maple (software)|Maple]] and [[Magma (software)|Magma]] these functions are named ''HilbertSeries'' and ''HilbertPolynomial''.
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| == References ==
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| * {{Citation| last=Eisenbud | first=David | author-link=David Eisenbud | year=1995 | title=Commutative algebra. With a view toward algebraic geometry | volume=150 | series=Graduate Texts in Mathematics | place=New York | publisher=Springer-Verlag | id={{MathSciNet|id=1322960}} | isbn=0-387-94268-8}}.
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| * {{Citation| last=Schenck | first=Hal | title=Computational Algebraic Geometry | publisher=[[Cambridge University Press]] | location=[[Cambridge]] | isbn=978-0-521-53650-9 | id={{MathSciNet|id=011360}} | year=2003}}
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| * {{Citation| last=Stanley | first=Richard | author-link=Richard P. Stanley | year=1978 | title=Hilbert functions of graded algebras | periodical=Advances in Math. | volume=28 | issue=1 | pages=57–83 | id={{MathSciNet|id=0485835}}| doi=10.1016/0001-8708(78)90045-2 }}.
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| [[Category:Commutative algebra]]
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| [[Category:Algebraic geometry]]
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