View source for Polarization of an algebraic form

{{short description|Technique for expressing a homogeneous polynomial in a simpler fashion by adjoining more variables}}
{{About|formulas for higher-degree polynomials|formula that relates norms to inner products|Polarization identity}}
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 unique symmetric [[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 <math>f(\mathbf{u})</math> be a polynomial in <math>n</math> variables <math>\mathbf{u} = \left(u_1, u_2, \ldots, u_n\right).</math> Suppose that <math>f</math> is homogeneous of degree <math>d,</math> which means that 
<math display="block>f(t \mathbf{u}) = t^d f(\mathbf{u}) \quad \text{ for all } t.</math>

Let <math>\mathbf{u}^{(1)}, \mathbf{u}^{(2)}, \ldots, \mathbf{u}^{(d)}</math> be a collection of [[indeterminate (variable)|indeterminate]]s with <math>\mathbf{u}^{(i)} = \left(u^{(i)}_1, u^{(i)}_2, \ldots, u^{(i)}_n\right),</math> so that there are <math>d n</math> variables altogether.  The '''polar form''' of <math>f</math> is a polynomial 
<math display="block>F\left(\mathbf{u}^{(1)}, \mathbf{u}^{(2)}, \ldots, \mathbf{u}^{(d)}\right)</math>
which is linear separately in each <math>\mathbf{u}^{(i)}</math> (that is, <math>F</math> is multilinear), symmetric in the <math>\mathbf{u}^{(i)},</math> and such that
<math display="block>F\left(\mathbf{u}, \mathbf{u}, \ldots, \mathbf{u}\right) = f(\mathbf{u}).</math>

The polar form of <math>f</math> is given by the following construction
<math display="block>F\left({\mathbf u}^{(1)}, \dots, {\mathbf u}^{(d)}\right) = \frac{1}{d!}\frac{\partial}{\partial\lambda_1} \dots \frac{\partial}{\partial\lambda_d}f(\lambda_1{\mathbf u}^{(1)} + \dots + \lambda_d{\mathbf u}^{(d)})|_{\lambda=0}.</math>
In other words, <math>F</math> is a constant multiple of the coefficient of <math>\lambda_1 \lambda_2 \ldots \lambda_d</math> in the expansion of <math>f\left(\lambda_1 \mathbf{u}^{(1)} + \cdots + \lambda_d \mathbf{u}^{(d)}\right).</math>

==Examples==

'''A quadratic example.'''  Suppose that <math>\mathbf{x} = (x, y)</math> and <math>f(\mathbf{x})</math> is the [[quadratic form]]
<math display="block">f(\mathbf{x}) = x^2 + 3 x y + 2 y^2.</math>
Then the polarization of <math>f</math> is a function in <math>\mathbf{x}^{(1)} = \left(x^{(1)}, y^{(1)}\right)</math> and <math>\mathbf{x}^{(2)} = \left(x^{(2)}, y^{(2)}\right)</math> given by
<math display="block">F\left(\mathbf{x}^{(1)}, \mathbf{x}^{(2)}\right) = x^{(1)} x^{(2)} + \frac{3}{2} x^{(2)} y^{(1)} + \frac{3}{2} x^{(1)} y^{(2)} + 2 y^{(1)} y^{(2)}.</math>
More generally, if <math>f</math> is any quadratic form then the polarization of <math>f</math> agrees with the conclusion of the [[polarization identity]].

'''A cubic example.'''  Let <math>f(x, y) = x^3 + 2xy^2.</math> Then the polarization of <math>f</math> is given by
<math display="block">F\left(x^{(1)}, y^{(1)}, x^{(2)}, y^{(2)}, x^{(3)}, y^{(3)}\right) = 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>

==Mathematical details and consequences==

The polarization of a homogeneous polynomial of degree <math>d</math> is valid over any [[commutative ring]] in which <math>d!</math> is a unit.  In particular, it holds over any [[Field (mathematics)|field]] of [[characteristic zero]] or whose characteristic is strictly greater than <math>d.</math>

===The polarization isomorphism (by degree)===

For simplicity, let <math>k</math> be a field of characteristic zero and let <math>A = k[\mathbf{x}]</math> be the [[polynomial ring]] in <math>n</math> variables over <math>k.</math>  Then <math>A</math> is [[graded ring|graded]] by [[degree of a polynomial|degree]], so that
<math display="block">A = \bigoplus_d A_d.</math>
The polarization of algebraic forms then induces an isomorphism of vector spaces in each degree
<math display="block">A_d \cong \operatorname{Sym}^d k^n</math>
where <math>\operatorname{Sym}^d</math> is the <math>d</math>-th [[symmetric power]] of the <math>n</math>-dimensional space <math>k^n.</math> 

These isomorphisms can be expressed independently of a basis as follows.  If <math>V</math> is a finite-dimensional vector space and <math>A</math> is the ring of <math>k</math>-valued polynomial functions on <math>V</math> graded by homogeneous degree, then polarization yields an isomorphism
<math display="block">A_d \cong \operatorname{Sym}^d V^*.</math>

===The algebraic isomorphism===

Furthermore, the polarization is compatible with the algebraic structure on <math>A</math>, so that
<math display="block">A \cong \operatorname{Sym}^{\bullet} V^*</math>
where <math>\operatorname{Sym}^{\bullet} V^*</math> is the full [[symmetric algebra]] over <math>V^*.</math>

===Remarks===

* For fields of [[positive characteristic]] <math>p,</math> the foregoing isomorphisms apply if the graded algebras are truncated at degree <math>p - 1.</math>
* There do exist generalizations when <math>V</math> is an infinite dimensional [[topological vector space]].

==See also==

* {{annotated link|Homogeneous function}}

==References==
* [[Claudio Procesi]] (2007) ''Lie Groups: an approach through invariants and representations'', Springer, {{isbn|9780387260402}} .

{{DEFAULTSORT:Polarization Of An Algebraic Form}}
[[Category:Abstract algebra]]
[[Category:Homogeneous polynomials]]