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 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 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({{\mathbf u}}^{{(1)}},\dots ,{{\mathbf u}}^{{(d)}})={\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}}.

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({{\mathbf x}})=x^{2}+3xy+2y^{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({{\mathbf x}}^{{(1)}},{{\mathbf x}}^{{(2)}})=x^{{(1)}}x^{{(2)}}+{\frac {3}{2}}x^{{(2)}}y^{{(1)}}+{\frac {3}{2}}x^{{(1)}}y^{{(2)}}+2y^{{(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)}}+{\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)}}.

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.

References

Claudio Procesi (2007) Lie Groups: an approach through invariants and representation, Springer, ISBN 9780387260402 .

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