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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Hilbert polynomial ： ウィキペディア英語版 Hilbert series and Hilbert polynomial Given a graded commutative algebra finitely generated over a field, the Hilbert function, Hilbert polynomial, and Hilbert series are three strongly related notions which measure the growth of the dimension of its homogeneous components. These notions have been extended to filtered algebras and graded filtered modules over these algebras. The typical situations where these notions are used are the following: * The quotient by a homogeneous ideal of a multivariate polynomial ring, graded by the total degree. * The quotient by an ideal of a multivariate polynomial ring, filtered by the total degree. * The filtration of a local ring by the powers of its maximal ideal. In this case the Hilbert polynomial is called the Hilbert–Samuel polynomial. The Hilbert series of an algebra or a module is a special case of the Hilbert–Poincaré series of a graded vector space. 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. == Definitions and main properties== Let us consider a finitely generated graded commutative algebra ''S'' over a field ''K'', which is finitely generated by elements of positive degree. This means that :$S\; =\; \backslash bigoplus\_\; S\_i\backslash$ and that $S\_0=K$. The Hilbert function :$HF\_S\backslash ;:\backslash ;n\backslash mapsto\; \backslash dim\_K\backslash ,S\_n$ maps the integer ''n'' onto the dimension of the ''K''-vector space ''S''''n''. The Hilbert series, which is called Hilbert–Poincaré series in the more general setting of graded vector spaces, is the formal series :$HS\_S(t)=\backslash sum\_^\; HF\_S(n)\backslash ,t^n.$ If ''S'' is generated by ''h'' homogeneous elements of positive degrees $d\_1,\; \backslash ldots,\; d\_h$, then the sum of the Hilbert series is a rational fraction :$HS\_S(t)=\backslash frac)\}\backslash ,,$ where ''Q'' is a polynomial with integer coefficients. If ''S'' is generated by elements of degree 1 then the sum of the Hilbert series may be rewritten as :$HS\_S(t)=\backslash frac\backslash ,,$ where ''P'' is a polynomial with positive integer coefficients. In this case the series expansion of this rational fraction is :$HS\_S(t)=P(t)\backslash ,\backslash left(1+\backslash delta\backslash ,t+\backslash cdots\; +\backslash binom\backslash ,t^n+\backslash cdots\backslash right)$ where the binomial coefficient $\backslash binom$ is $\backslash ;\backslash frac\backslash ;$ for $n>-\backslash delta$ and 0 otherwise. This shows that there exists a unique polynomial $HP\_S(n)$ with rational coefficients which is equal to $HF\_S(n)$ for $n\backslash ge\; \backslash deg\; P-\backslash delta+1$. This polynomial is the Hilbert polynomial. The least ''n''0 such that $HP\_S(n)=HF\_S(n)$ for ''n'' ≥ ''n''0 is called the Hilbert regularity. It may be lower than $\backslash deg\; P-\backslash delta+1$. The Hilbert polynomial is a numerical polynomial, since the dimensions are integers, but the polynomial almost never has integer coefficients . All these definitions may be extended to finitely generated graded modules over ''S'', with the only difference that a factor ''t''''m'' appears in the Hilbert series, where ''m'' is the minimal degree of the generators of the module, which may be negative. The Hilbert function, the Hilbert series and the Hilbert polynomial of a filtered algebra are those of the associated graded algebra. The Hilbert polynomial of a projective variety ''V'' in P''n'' is defined as the Hilbert polynomial of the homogeneous coordinate ring of ''V''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Hilbert series and Hilbert polynomial」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.