Polarization_of_an_algebraic

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Polarization of an algebraic form

Summary

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 unique symmetric 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 techniqueEdit

The fundamental ideas are as follows. Let $f(\mathbf {u} )$ be a polynomial in $n$ variables $\mathbf {u} =\left(u_{1},u_{2},\ldots ,u_{n}\right).$ Suppose that $f$ is homogeneous of degree $d,$ which means that

f(t\mathbf {u} )=t^{d}f(\mathbf {u} )\quad {\text{ for all }}t.

Let $\mathbf {u} ^{(1)},\mathbf {u} ^{(2)},\ldots ,\mathbf {u} ^{(d)}$ be a collection of indeterminates with $\mathbf {u} ^{(i)}=\left(u_{1}^{(i)},u_{2}^{(i)},\ldots ,u_{n}^{(i)}\right),$ so that there are $dn$ variables altogether. The polar form of $f$ is a polynomial

F\left(\mathbf {u} ^{(1)},\mathbf {u} ^{(2)},\ldots ,\mathbf {u} ^{(d)}\right)

which is linear separately in each

\mathbf {u} ^{(i)}

(that is,

F

is multilinear), symmetric in the

\mathbf {u} ^{(i)},

and such that

F\left(\mathbf {u} ,\mathbf {u} ,\ldots ,\mathbf {u} \right)=f(\mathbf {u} ).

The polar form of $f$ is given by the following construction

F\left({\mathbf {u} }^{(1)},\dots ,{\mathbf {u} }^{(d)}\right)={\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

\lambda _{1}\lambda _{2}\ldots \lambda _{d}

in the expansion of

f\left(\lambda _{1}\mathbf {u} ^{(1)}+\cdots +\lambda _{d}\mathbf {u} ^{(d)}\right).

ExamplesEdit

A quadratic example. Suppose that $\mathbf {x} =(x,y)$ and $f(\mathbf {x} )$ is the quadratic form

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

Then the polarization of

f

is a function in

\mathbf {x} ^{(1)}=\left(x^{(1)},y^{(1)}\right)

and

\mathbf {x} ^{(2)}=\left(x^{(2)},y^{(2)}\right)

given by

F\left(\mathbf {x} ^{(1)},\mathbf {x} ^{(2)}\right)=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 $f(x,y)=x^{3}+2xy^{2}.$ Then the polarization of $f$ is given by

F\left(x^{(1)},y^{(1)},x^{(2)},y^{(2)},x^{(3)},y^{(3)}\right)=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 consequencesEdit

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 $A=k[\mathbf {x} ]$ 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 \operatorname {Sym} ^{d}k^{n}

where

\operatorname {Sym} ^{d}

is the

d

-th symmetric power of the

n

-dimensional space

k^{n}.

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 \operatorname {Sym} ^{d}V^{*}.

The algebraic isomorphismEdit

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

A\cong \operatorname {Sym} ^{\cdot }V^{*}

where

\operatorname {Sym} ^{\cdot }V^{*}

is the full symmetric algebra over

V^{*}.

RemarksEdit

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.

ReferencesEdit

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