Polarization of an algebraic form

The technique

The fundamental ideas are as follows. Let f(u) be a polynomial in 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 indeterminates with u⁽ⁱ⁾ = (u₁⁽ⁱ⁾, u₂⁽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

F ( u ( 1 ) , … , u ( d ) ) = 1 d ! ∂ ∂ λ 1 … ∂ ∂ λ d f ( λ 1 u ( 1 ) + ⋯ + λ d u ( d ) ) | λ = 0 .

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 ) = 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 ( 1 ) , x ( 2 ) ) = x ( 1 ) x ( 2 ) + 3 2 x ( 2 ) y ( 1 ) + 3 2 x ( 1 ) y ( 2 ) + 2 y ( 1 ) y ( 2 ) .

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 ( 1 ) , y ( 1 ) , x ( 2 ) , y ( 2 ) , x ( 3 ) , y ( 3 ) ) = x ( 1 ) x ( 2 ) x ( 3 ) + 2 3 x ( 1 ) y ( 2 ) y ( 3 ) + 2 3 x ( 3 ) y ( 1 ) y ( 2 ) + 2 3 x ( 2 ) y ( 3 ) y ( 1 ) .

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 = ⨁ d A d .

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

A d ≅ S y m 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 ≅ S y m d V ∗ .

The algebraic isomorphism

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

A ≅ S y m ⋅ 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.