|
In mathematics, in particular in algebra, polarization is a technique for expressing a homogeneous polynomial in a simpler fashion by adjoining more variables. Specifically, given a homogeneous polynomial, polarization produces a multilinear form from which the original polynomial can be recovered by evaluating along a certain diagonal. Although the technique is deceptively simple, it has applications in many areas of abstract mathematics: in particular to algebraic geometry, invariant theory, and representation theory. Polarization and related techniques form the foundations for Weyl's invariant theory. ==The technique== The fundamental ideas are as follows. Let ''f''(u) be a polynomial in ''n'' variables u = (''u''1, ''u''2, ..., ''u''n). Suppose that ''f'' is homogeneous of degree ''d'', which means that :''f''(''t'' u) = ''t''''d'' ''f''(u) for all ''t''. Let u(1), u(2), ..., u(d) be a collection of indeterminates with u(i) = (''u''1(i), ''u''2(i), ..., ''u''n(i)), so that there are ''dn'' variables altogether. The polar form of ''f'' is a polynomial :''F''(u(1), u(2), ..., u(d)) which is linear separately in each u(i) (i.e., ''F'' is multilinear), symmetric in the u(i), and such that :''F''(u,u, ..., u)=''f''(u). The polar form of ''f'' is given by the following construction : In other words, ''F'' is a constant multiple of the coefficient of λ1 λ2...λd in the expansion of ''f''(λ1u(1) + ... + λdu(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「Polarization of an algebraic form」の詳細全文を読む スポンサード リンク
|