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''of a graded module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047310/h0473101.png" />''
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''of a [[graded module]] $M = \bigoplus_n M_n$''
  
A polynomial expressing the dimensions of the homogeneous components of the module as a function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047310/h0473102.png" /> for large natural numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047310/h0473103.png" />. More exactly, the following theorem, demonstrated in essence by D. Hilbert, is valid. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047310/h0473104.png" /> be a ring of polynomials over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047310/h0473105.png" />, graded so that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047310/h0473106.png" /> are homogeneous elements of the first degree, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047310/h0473107.png" /> be a graded <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047310/h0473108.png" />-module of finite type; then there exists a polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047310/h0473109.png" /> with rational coefficients such that, for sufficiently large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047310/h04731010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047310/h04731011.png" />. This polynomial is called the Hilbert polynomial.
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A polynomial expressing the dimensions of the homogeneous components of the module as a function of $n$ for large natural numbers $n$. More exactly, the following theorem, demonstrated in essence by D. Hilbert, is valid. Let $A = K[X_0,\ldots,X_m]$ be a ring of polynomials over a field $K$, graded so that the $X_i$ are homogeneous elements of the first degree, and let $M = \bigoplus_n M_n$ be a graded $A$-module of finite type; the ''Hilbert function'' of $M$ is $h_M(n) = \dim_K M_n$ and  there exists a polynomial $P_M(t)$ with rational coefficients such that, for sufficiently large $n$, $P_M(n) = h_M(n) = \dim_K M_n$. This polynomial is called the ''Hilbert polynomial''.
  
Of greatest interest is the interpretation of the Hilbert polynomial of a graded ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047310/h04731012.png" /> which is the quotient ring of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047310/h04731013.png" /> by a homogeneous ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047310/h04731014.png" />; in such a case the Hilbert polynomial gives the projective invariants of the projective variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047310/h04731015.png" /> defined by the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047310/h04731016.png" />. In particular, the degree of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047310/h04731017.png" /> coincides with the dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047310/h04731018.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047310/h04731019.png" /> is said to be the [[Arithmetic genus|arithmetic genus]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047310/h04731020.png" />. Hilbert polynomials also serve to express the degree of the imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047310/h04731021.png" />. The Hilbert polynomial of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047310/h04731022.png" /> is also the name given to the Hilbert polynomial of the projective variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047310/h04731023.png" /> with respect to the imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047310/h04731024.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047310/h04731025.png" /> is the invertible sheaf corresponding to this imbedding, then
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Of greatest interest is the interpretation of the Hilbert polynomial of a [[graded algebra]] $R$ which is the quotient ring of the ring $A$ by a homogeneous ideal $I$; in such a case the Hilbert polynomial gives the projective invariants of the projective variety $X = \text{Proj}(R) \subset \mathbf{P}^m$ defined by the ideal $I$. In particular, the degree of $P_R(t)$ coincides with the dimension of $X$, while $P_{\mathrm{A}} = (-1)^{\dim X}(P_R(0)-1)$ is said to be the ''[[arithmetic genus]]'' of $X$. Hilbert polynomials also serve to express the degree of the imbedding $X \subset \mathbf{P}^m$. The Hilbert polynomial of the ring $R$ is also the name given to the Hilbert polynomial of the projective variety $X$ with respect to the imbedding $X \subset \mathbf{P}^m$. If $\mathcal{O}_X(1)$ is the invertible sheaf corresponding to this imbedding, then
 +
$$
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P_R(n) = \dim_K H^0(X,\mathcal{O}_X(1)^{{\otimes} n})
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$$
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for sufficiently large $n$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047310/h04731026.png" /></td> </tr></table>
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====References====
 +
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  D. Hilbert,  "Gesammelte Abhandlungen" , '''2''' , Springer  (1933)</TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top"> M. Baldassarri,  "Algebraic varieties" , Springer  (1956)</TD></TR>
 +
<TR><TD valign="top">[3]</TD> <TD valign="top">  O. Zariski,  P. Samuel,  "Commutative algebra" , '''1''' , Springer  (1975)</TD></TR>
 +
</table>
  
for sufficiently large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047310/h04731027.png" />.
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{{TEX|done}}
 
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  D. Hilbert,  "Gesammelte Abhandlungen" , '''2''' , Springer  (1933)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M. Baldassarri,  "Algebraic varieties" , Springer  (1956)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  O. Zariski,  P. Samuel,  "Commutative algebra" , '''1''' , Springer  (1975)</TD></TR></table>
 

Revision as of 20:21, 21 August 2016

of a graded module $M = \bigoplus_n M_n$

A polynomial expressing the dimensions of the homogeneous components of the module as a function of $n$ for large natural numbers $n$. More exactly, the following theorem, demonstrated in essence by D. Hilbert, is valid. Let $A = K[X_0,\ldots,X_m]$ be a ring of polynomials over a field $K$, graded so that the $X_i$ are homogeneous elements of the first degree, and let $M = \bigoplus_n M_n$ be a graded $A$-module of finite type; the Hilbert function of $M$ is $h_M(n) = \dim_K M_n$ and there exists a polynomial $P_M(t)$ with rational coefficients such that, for sufficiently large $n$, $P_M(n) = h_M(n) = \dim_K M_n$. This polynomial is called the Hilbert polynomial.

Of greatest interest is the interpretation of the Hilbert polynomial of a graded algebra $R$ which is the quotient ring of the ring $A$ by a homogeneous ideal $I$; in such a case the Hilbert polynomial gives the projective invariants of the projective variety $X = \text{Proj}(R) \subset \mathbf{P}^m$ defined by the ideal $I$. In particular, the degree of $P_R(t)$ coincides with the dimension of $X$, while $P_{\mathrm{A}} = (-1)^{\dim X}(P_R(0)-1)$ is said to be the arithmetic genus of $X$. Hilbert polynomials also serve to express the degree of the imbedding $X \subset \mathbf{P}^m$. The Hilbert polynomial of the ring $R$ is also the name given to the Hilbert polynomial of the projective variety $X$ with respect to the imbedding $X \subset \mathbf{P}^m$. If $\mathcal{O}_X(1)$ is the invertible sheaf corresponding to this imbedding, then $$ P_R(n) = \dim_K H^0(X,\mathcal{O}_X(1)^{{\otimes} n}) $$ for sufficiently large $n$.

References

[1] D. Hilbert, "Gesammelte Abhandlungen" , 2 , Springer (1933)
[2] M. Baldassarri, "Algebraic varieties" , Springer (1956)
[3] O. Zariski, P. Samuel, "Commutative algebra" , 1 , Springer (1975)
How to Cite This Entry:
Hilbert polynomial. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hilbert_polynomial&oldid=39065
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article