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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 Quasi-homogeneous polynomial ： ウィキペディア英語版 Quasi-homogeneous polynomial In algebra, a multivariate polynomial : $f(x)=\backslash sum\_\backslash alpha\; a\_\backslash alpha\; x^\backslash alpha\backslash text\backslash alpha=(i\_1,\backslash dots,i\_r)\backslash in\; \backslash mathbb^r\; \backslash text\; x^\backslash alpha=x\_1^\; \backslash cdots\; x\_r^,$ is quasi-homogeneous or weighted homogeneous, if there exists ''r'' integers $w\_1,\; \backslash ldots,\; w\_r$, called weights of the variables, such that the sum $w=w\_1i\_1+\; \backslash cdots\; +\; w\_ri\_r$ is the same for all nonzero terms of ''f''. This sum ''w'' is the ''weight'' or the ''degree'' of the polynomial. The term ''quasi-homogeneous'' comes form the fact that a polynomial ''f'' is quasi-homogeneous if and only if :$f(\backslash lambda^\; x\_1,\; \backslash ldots,\; \backslash lambda^\; x\_r)=\backslash lambda^w\; f(x\_1,\backslash ldots,\; x\_r)$ for every $\backslash lambda$ in any field containing the coefficients. A polynomial $f(x\_1,\; \backslash ldots,\; x\_n)$ is quasi-homogeneous with weights $w\_1,\; \backslash ldots,\; w\_r$ if and only if :$f(y\_1^,\; \backslash ldots,\; y\_n^)$ is a homogeneous polynomial in the $y\_i$. In particular, a homogeneous polynomial is always quasi-homogeneous, with all weights equal to 1. In other words, a polynomial is quasi-homogeneous if all the $\backslash alpha$ belong to the same affine hyperplane. As the Newton polygon of the polynomial is the convex hull of the set $\backslash ,$ the quasi-homogeneous polynomials may also be defined as the polynomials that have a degenerate Newton polynomial (here "degenerate" means "contained in some affine hyperplane"). ==Introduction== Consider the polynomial $f(x,y)=5x^3y^3+xy^9-2y^$. This one has no chance of being a homogeneous polynomial; however if instead of considering $f(\backslash lambda\; x,\backslash lambda\; y)$ we use the pair $(\backslash lambda^3,\backslash lambda)$ to test ''homogeneity'', then : $f(\backslash lambda^3\; x,\backslash lambda\; y)=5(\backslash lambda^3x)^3(\backslash lambda\; y)^3+(\backslash lambda^3x)(\backslash lambda\; y)^9-2(\backslash lambda\; y)^=\backslash lambda^f(x,y).\; \backslash ,$ We say that $f(x,y)$ is a quasi-homogeneous polynomial of type (3,1), because its three pairs (''i''1,''i''2) of exponents (3,3), (1,9) and (0,12) all satisfy the linear equation $3i\_1+1i\_2=12$. In particular, this says that the Newton polygon of $f(x,y)$ lies in the affine space with equation $3x+y=12$ inside $\backslash mathbb^2$. The above equation is equivalent to this new one: $\backslash tfracx+\backslash tfracy=1$. Some authors〔J. Steenbrink (1977). ''Compositio Mathematica'', tome 34, n° 2. Noordhoff International Publishing. p. 211 (Available on-line at (Numdam ))〕 prefer to use this last condition and prefer to say that our polynomial is quasi-homogeneous of type ($\backslash tfrac,\backslash tfrac$). As noted above, a homogeneous polynomial $g(x,y)$ of degree ''d'' is just a quasi-homogeneous polynomial of type (1,1); in this case all its pairs of exponents will satisfy the equation $1i\_1+1i\_2=d$. 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「Quasi-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.