Polarization of an algebraic form

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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.

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.

Contents

1 The Technique
2 Examples
3 Mathematical details and consequences

3.1 The polarization isomorphism (by degree)
3.2 The algebraic isomorphism
3.3 Remarks

The Technique

The fundamental ideas are as follows. Let f(u) be a polynomial of n variables u = (u₁, u₂, ..., 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 indeterminants with u⁽ⁱ⁾ = (u₁⁽ⁱ⁾, u₂⁽ⁱ⁾, ..., u_n⁽ⁱ⁾), so that there are d*n 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) and such that

F(u,u, ..., u)=f(u).

The polar form of f is given by the following construction

[F(^,\dots,^)=\frac\frac\dots\fracf(\lambda_1^+\dots+\lambda_d^).]

In other words, F is a constant multiple of the coefficient of λ₁ λ₂...λ_d in the expansion of f(λ₁u⁽¹⁾ + ... + λ_du^(d)).

Examples

Suppose that x=(x,y) and f(x) is the quadratic form

[f() = x^2 + 3 x y + 2 y^2].

Then the polarization of f is a function in x⁽¹⁾ = (x⁽¹⁾, y⁽¹⁾) and x⁽²⁾ = (x⁽²⁾, y⁽²⁾) given by

[F(^,^) = x^x^+\fracx^y^+\fracx^y^+2 y^y^]

More generally, if f is any quadratic form, then the polarization of f agrees with the conclusion of the polarization identity.

A cubic example. Let f(x,y)=x³ + 2xy². Then the polarization of f is given by

[F(x^,y^,x^,y^,x^,y^)= x^x^x^+\fracx^y^y^+\fracx^y^y^+\fracx^y^y^.]

Mathematical details and consequences

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 of characteristic zero or whose characteristic is strictly greater than d.

The polarization isomorphism (by degree)

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 by degree, so that

[A = \bigoplus_d A_d.]

The polarization of algebraic forms then induces an isomorphism of vector spaces in each degree

[A_d \cong Sym^d k^n]

where Sym^d is the d-th symmetric power of the n-dimensional space kⁿ.

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

[A_d \cong Sym^d V^*.]

The algebraic isomorphism

Furthermore, the polarization is compatible with the algebraic structure on A, so that

[A \cong Sym^\cdot V^*]

where Sym^.V^* is the full symmetric algebra over V^*.

Remarks

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

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