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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 Homogenization of a polynomial ： ウィキペディア英語版 Homogeneous polynomial In mathematics, a homogeneous polynomial is a polynomial whose nonzero terms all have the same degree.〔D. Cox, J. Little, D. O'Shea: ''Using Algebraic Geometry'', 2nd ed., page 2. Springer-Verlag, 2005.〕 For example, $x^5\; +\; 2\; x^3\; y^2\; +\; 9\; x\; y^4$ is a homogeneous polynomial of degree 5, in two variables; the sum of the exponents in each term is always 5. The polynomial $x^3\; +\; 3\; x^2\; y\; +\; z^7$ is not homogeneous, because the sum of exponents does not match from term to term. A polynomial is homogeneous if and only if it defines a homogeneous function. An algebraic form, or simply form, is a function defined by a homogeneous polynomial.〔However, as some authors do not make a clear distinction between a polynomial and its associated function, the terms ''homogeneous polynomial'' and ''form'' are sometimes considered as synonymous.〕 A binary form is a form in two variables. A ''form'' is also a function defined on a vector space, which may be expressed as a homogeneous function of the coordinates over any basis. A polynomial of degree 0 is always homogeneous; it is simply an element of the field or ring of the coefficients, usually called a constant or a scalar. A form of degree 1 is a linear form.〔''Linear forms'' are defined only for finite-dimensional vector space, and have thus to be distinguished from ''linear functionals'', which are defined for every vector space. "Linear functional" is rarely used for finite-dimensional vector spaces.〕 A form of degree 2 is a quadratic form. In geometry, the Euclidean distance is the square root of a quadratic form. Homogeneous polynomials are ubiquitous in mathematics and physics.〔Homogeneous polynomials in physics often appear as a consequence of dimensional analysis, where measured quantities must match in real-world problems.〕 They play a fundamental role in algebraic geometry, as a projective algebraic variety is defined as the set of the common zeros of a set of homogeneous polynomials. ==Properties== A homogeneous polynomial defines a homogeneous function. This means that a multivariate polynomial ''P'' is homogeneous of degree ''d'' if and only if :$P(\backslash lambda\; x\_1,\; \backslash ldots,\; \backslash lambda\; x\_n)=\backslash lambda^d\backslash ,P(x\_1,\backslash ldots,x\_n)\backslash ,,$ for every $\backslash lambda$ in the field of the coefficients. In particular, if ''P'' is homogeneous then :$P(x\_1,\backslash ldots,x\_n)=0\; \backslash quad\backslash Rightarrow\backslash quad\; P(\backslash lambda\; x\_1,\; \backslash ldots,\; \backslash lambda\; x\_n)=0,$ for every $\backslash lambda.$ This property is fundamental in the definition of a projective variety. Any nonzero polynomial may be decomposed, in a unique way, as a sum of homogeneous polynomials of different degrees, which are called the homogeneous components of the polynomial. Given a polynomial ring $R=K(\backslash ldots,x\_n)$ over a field (or, more generally, a ring) ''K'', the homogeneous polynomials of degree ''d'' form a vector space (or a module), commonly denoted $R\_d.$ The above unique decomposition means that $R$ is the direct sum of the $R\_d$ (sum over all nonnegative integers). The dimension of the vector space (or free module) $R\_d$ is the number of different monomials of degree ''d'' in ''n'' variables (that is the maximal number of nonzero terms in a homogeneous polynomial of degree ''d'' in ''n'' variables). It is equal to the binomial coefficient :$\backslash binom=\backslash binom=\backslash frac.$ 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「Homogeneous 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.