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.

The technique[edit]

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) = td f(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

F({\bold u}^{(1)},\dots,{\bold u}^{(d)})=\frac{1}{d!}\frac{\partial}{\partial\lambda_1}\dots\frac{\partial}{\partial\lambda_d}f(\lambda_1{\bold u}^{(1)}+\dots+\lambda_d{\bold u}^{(d)})|_{\lambda=0}.

In other words, F is a constant multiple of the coefficient of λ1 λ2...λd in the expansion of f1u(1) + ... + λdu(d)).

Examples[edit]

f({\bold 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({\bold x}^{(1)},{\bold x}^{(2)}) = x^{(1)}x^{(2)}+\frac{3}{2}x^{(2)}y^{(1)}+\frac{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)}+\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[edit]

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)[edit]

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 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

A_d \cong Sym^d V^*.

The algebraic isomorphism[edit]

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[edit]

References[edit]