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| 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.
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| 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]].
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| ==The technique==
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| The fundamental ideas are as follows. Let ''f''('''u''') be a polynomial in ''n'' variables '''u''' = (''u''<sub>1</sub>, ''u''<sub>2</sub>, ..., ''u''<sub>n</sub>). Suppose that ''f'' is homogeneous of degree ''d'', which means that
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| :''f''(''t'' '''u''') = ''t''<sup>''d''</sup> ''f''('''u''') for all ''t''.
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| Let '''u'''<sup>(1)</sup>, '''u'''<sup>(2)</sup>, ..., '''u'''<sup>(d)</sup> be a collection of [[indeterminate]]s with '''u'''<sup>(i)</sup> = (''u''<sub>1</sub><sup>(i)</sup>, ''u''<sub>2</sub><sup>(i)</sup>, ..., ''u''<sub>n</sub><sup>(i)</sup>), so that there are ''dn'' variables altogether. The '''polar form''' of ''f'' is a polynomial
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| :''F''('''u'''<sup>(1)</sup>, '''u'''<sup>(2)</sup>, ..., '''u'''<sup>(d)</sup>)
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| which is linear separately in each '''u'''<sup>(i)</sup> (i.e., ''F'' is multilinear), symmetric in the '''u'''<sup>(i)</sup>, and such that
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| :''F''('''u''','''u''', ..., '''u''')=''f''('''u''').
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| The polar form of ''f'' is given by the following construction
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| :<math>F({\bold u}^{(1)},\dots,{\bold u}^{(d)})=\frac{1}{d!}\frac{\partial}{\partial\lambda_1}\dots\frac{\partial}{\partial\lambda_d}f(\lambda_1{\bold u}^{(1)}+\dots+\lambda_d{\bold u}^{(d)})|_{\lambda=0}.</math>
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| In other words, ''F'' is a constant multiple of the coefficient of λ<sub>1</sub> λ<sub>2</sub>...λ<sub>d</sub> in the expansion of ''f''(λ<sub>1</sub>'''u'''<sup>(1)</sup> + ... + λ<sub>d</sub>'''u'''<sup>(d)</sup>).
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| ==Examples==
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| *Suppose that '''x'''=(''x'',''y'') and ''f''('''x''') is the [[quadratic form]]
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| :<math>f({\bold x}) = x^2 + 3 x y + 2 y^2.</math> | |
| Then the polarization of ''f'' is a function in '''x'''<sup>(1)</sup> = (''x''<sup>(1)</sup>, ''y''<sup>(1)</sup>) and '''x'''<sup>(2)</sup> = (''x''<sup>(2)</sup>, ''y''<sup>(2)</sup>) given by
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| :<math>F({\bold x}^{(1)},{\bold x}^{(2)}) = x^{(1)}x^{(2)}+\frac{3}{2}x^{(2)}y^{(1)}+\frac{3}{2}x^{(1)}y^{(2)}+2 y^{(1)}y^{(2)}.</math>
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| *More generally, if ''f'' is any quadratic form, then the polarization of ''f'' agrees with the conclusion of the [[polarization identity]].
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| *'''A cubic example.''' Let ''f''(''x'',''y'')=''x''<sup>3</sup> + 2''xy''<sup>2</sup>. Then the polarization of ''f'' is given by
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| :<math>F(x^{(1)},y^{(1)},x^{(2)},y^{(2)},x^{(3)},y^{(3)})= x^{(1)}x^{(2)}x^{(3)}+\frac{2}{3}x^{(1)}y^{(2)}y^{(3)}+\frac{2}{3}x^{(3)}y^{(1)}y^{(2)}+\frac{2}{3}x^{(2)}y^{(3)}y^{(1)}.</math>
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| ==Mathematical details and consequences==
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| The polarization of a homogeneous polynomial of degree ''d'' is valid over any [[commutative ring]] in which ''d''! is a unit. In particular, it holds over any [[Field (mathematics)|field]] of [[characteristic zero]] or whose characteristic is strictly greater than ''d''.
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| ===The polarization isomorphism (by degree)===
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| For simplicity, let ''k'' be a field of characteristic zero and let ''A''=''k''['''x'''] be the [[polynomial ring]] in ''n'' variables over ''k''. Then ''A'' is [[graded ring|graded]] by [[degree of a polynomial|degree]], so that
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| :<math>A = \bigoplus_d A_d.</math>
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| The polarization of algebraic forms then induces an isomorphism of vector spaces in each degree
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| :<math>A_d \cong Sym^d k^n</math> | |
| where ''Sym''<sup>''d''</sup> is the ''d''-th [[symmetric power]] of the ''n''-dimensional space ''k''<sup>n</sup>.
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| These isomorphisms can be expressed independently of a basis as follows. If ''V'' is a finite-dimensional vector space and ''A'' is the ring of ''k''-valued polynomial functions on ''V'', graded by homogeneous degree, then polarization yields an isomorphism
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| :<math>A_d \cong Sym^d V^*.</math>
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| ===The algebraic isomorphism===
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| Furthermore, the polarization is compatible with the algebraic structure on ''A'', so that
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| :<math>A \cong Sym^\cdot V^*</math>
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| where ''Sym''<sup>.</sup>''V''<sup>*</sup> is the full [[symmetric algebra]] over ''V''<sup>*</sup>.
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| ===Remarks===
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| * For fields of [[positive characteristic]] ''p'', the foregoing isomorphisms apply if the graded algebras are truncated at degree ''p''-1.
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| * There do exist generalizations when ''V'' is an infinite dimensional [[topological vector space]].
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| ==References==
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| * [[Claudio Procesi]] (2007) ''Lie Groups: an approach through invariants and representation'', Springer, ISBN 9780387260402 .
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| {{DEFAULTSORT:Polarization Of An Algebraic Form}}
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| [[Category:Abstract algebra]]
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| [[Category:Homogeneous polynomials]]
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She is recognized by the title of Myrtle Shryock. For years he's been living in North Dakota and his family members enjoys it. To gather coins is what his family and him enjoy. For many years I've been working as a payroll clerk.
Stop by my web blog ... http://srgame.co.kr/qna/21862