Polarization of An Algebraic Form
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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, ..., un). Suppose that f is homogeneous of degree d, which means that

f(t u) = tdf(u) for all t.

Let u(1), u(2), ..., u(d) be a collection of indeterminates with u(i) = (u1(i), u2(i), ..., un(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

${\displaystyle F({\mathbf {u} }^{(1)},\dots ,{\mathbf {u} }^{(d)})={\frac {1}{d!}}{\frac {\partial }{\partial \lambda _{1}}}\dots {\frac {\partial }{\partial \lambda _{d}}}f(\lambda _{1}{\mathbf {u} }^{(1)}+\dots +\lambda _{d}{\mathbf {u} }^{(d)})|_{\lambda =0}.}$

In other words, F is a constant multiple of the coefficient of ?1 ?2...?d in the expansion of f(?1u(1) + ... + ?du(d)).

## Examples

${\displaystyle f({\mathbf {x} })=x^{2}+3xy+2y^{2}.}$

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

${\displaystyle F({\mathbf {x} }^{(1)},{\mathbf {x} }^{(2)})=x^{(1)}x^{(2)}+{\frac {3}{2}}x^{(2)}y^{(1)}+{\frac {3}{2}}x^{(1)}y^{(2)}+2y^{(1)}y^{(2)}.}$
• 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
${\displaystyle F(x^{(1)},y^{(1)},x^{(2)},y^{(2)},x^{(3)},y^{(3)})=x^{(1)}x^{(2)}x^{(3)}+{\frac {2}{3}}x^{(1)}y^{(2)}y^{(3)}+{\frac {2}{3}}x^{(3)}y^{(1)}y^{(2)}+{\frac {2}{3}}x^{(2)}y^{(3)}y^{(1)}.}$

## 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 be the polynomial ring in n variables over k. Then A is graded by degree, so that

${\displaystyle A=\bigoplus _{d}A_{d}.}$

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

${\displaystyle A_{d}\cong Sym^{d}k^{n}}$

where Symd 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

${\displaystyle A_{d}\cong Sym^{d}V^{*}.}$

### The algebraic isomorphism

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

${\displaystyle A\cong Sym^{\cdot }V^{*}}$

where SymV* is the full symmetric algebra over V*.

## References

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