Polarization of an algebraic form
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]\displaystyle{ f(\mathbf{u}) }[/math] be a polynomial in [math]\displaystyle{ n }[/math] variables [math]\displaystyle{ \mathbf{u} = \left(u_1, u_2, \ldots, u_n\right). }[/math] Suppose that [math]\displaystyle{ f }[/math] is homogeneous of degree [math]\displaystyle{ d, }[/math] which means that [math]\displaystyle{ f(t \mathbf{u}) = t^d f(\mathbf{u}) \quad \text{ for all } t. }[/math]
Let [math]\displaystyle{ \mathbf{u}^{(1)}, \mathbf{u}^{(2)}, \ldots, \mathbf{u}^{(d)} }[/math] be a collection of indeterminates with [math]\displaystyle{ \mathbf{u}^{(i)} = \left(u^{(i)}_1, u^{(i)}_2, \ldots, u^{(i)}_n\right), }[/math] so that there are [math]\displaystyle{ d n }[/math] variables altogether. The polar form of [math]\displaystyle{ f }[/math] is a polynomial [math]\displaystyle{ F\left(\mathbf{u}^{(1)}, \mathbf{u}^{(2)}, \ldots, \mathbf{u}^{(d)}\right) }[/math] which is linear separately in each [math]\displaystyle{ \mathbf{u}^{(i)} }[/math] (that is, [math]\displaystyle{ F }[/math] is multilinear), symmetric in the [math]\displaystyle{ \mathbf{u}^{(i)}, }[/math] and such that [math]\displaystyle{ F\left(\mathbf{u}, \mathbf{u}, \ldots, \mathbf{u}\right) = f(\mathbf{u}). }[/math]
The polar form of [math]\displaystyle{ f }[/math] is given by the following construction [math]\displaystyle{ 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]\displaystyle{ F }[/math] is a constant multiple of the coefficient of [math]\displaystyle{ \lambda_1 \lambda_2 \ldots \lambda_d }[/math] in the expansion of [math]\displaystyle{ f\left(\lambda_1 \mathbf{u}^{(1)} + \cdots + \lambda_d \mathbf{u}^{(d)}\right). }[/math]
Examples
A quadratic example. Suppose that [math]\displaystyle{ \mathbf{x} = (x, y) }[/math] and [math]\displaystyle{ f(\mathbf{x}) }[/math] is the quadratic form [math]\displaystyle{ f(\mathbf{x}) = x^2 + 3 x y + 2 y^2. }[/math] Then the polarization of [math]\displaystyle{ f }[/math] is a function in [math]\displaystyle{ \mathbf{x}^{(1)} = \left(x^{(1)}, y^{(1)}\right) }[/math] and [math]\displaystyle{ \mathbf{x}^{(2)} = \left(x^{(2)}, y^{(2)}\right) }[/math] given by [math]\displaystyle{ 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]\displaystyle{ f }[/math] is any quadratic form then the polarization of [math]\displaystyle{ f }[/math] agrees with the conclusion of the polarization identity.
A cubic example. Let [math]\displaystyle{ f(x, y) = x^3 + 2xy^2. }[/math] Then the polarization of [math]\displaystyle{ f }[/math] is given by [math]\displaystyle{ 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]\displaystyle{ d }[/math] is valid over any commutative ring in which [math]\displaystyle{ d! }[/math] is a unit. In particular, it holds over any field of characteristic zero or whose characteristic is strictly greater than [math]\displaystyle{ d. }[/math]
The polarization isomorphism (by degree)
For simplicity, let [math]\displaystyle{ k }[/math] be a field of characteristic zero and let [math]\displaystyle{ A = k[\mathbf{x}] }[/math] be the polynomial ring in [math]\displaystyle{ n }[/math] variables over [math]\displaystyle{ k. }[/math] Then [math]\displaystyle{ A }[/math] is graded by degree, so that [math]\displaystyle{ A = \bigoplus_d A_d. }[/math] The polarization of algebraic forms then induces an isomorphism of vector spaces in each degree [math]\displaystyle{ A_d \cong \operatorname{Sym}^d k^n }[/math] where [math]\displaystyle{ \operatorname{Sym}^d }[/math] is the [math]\displaystyle{ d }[/math]-th symmetric power of the [math]\displaystyle{ n }[/math]-dimensional space [math]\displaystyle{ k^n. }[/math]
These isomorphisms can be expressed independently of a basis as follows. If [math]\displaystyle{ V }[/math] is a finite-dimensional vector space and [math]\displaystyle{ A }[/math] is the ring of [math]\displaystyle{ k }[/math]-valued polynomial functions on [math]\displaystyle{ V }[/math] graded by homogeneous degree, then polarization yields an isomorphism [math]\displaystyle{ A_d \cong \operatorname{Sym}^d V^*. }[/math]
The algebraic isomorphism
Furthermore, the polarization is compatible with the algebraic structure on [math]\displaystyle{ A }[/math], so that [math]\displaystyle{ A \cong \operatorname{Sym}^{\bullet} V^* }[/math] where [math]\displaystyle{ \operatorname{Sym}^{\bullet} V^* }[/math] is the full symmetric algebra over [math]\displaystyle{ V^*. }[/math]
Remarks
- For fields of positive characteristic [math]\displaystyle{ p, }[/math] the foregoing isomorphisms apply if the graded algebras are truncated at degree [math]\displaystyle{ p - 1. }[/math]
- There do exist generalizations when [math]\displaystyle{ V }[/math] is an infinite dimensional topological vector space.
See also
- Homogeneous function – Function with a multiplicative scaling behaviour
References
- Claudio Procesi (2007) Lie Groups: an approach through invariants and representations, Springer, ISBN 9780387260402 .
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