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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 technique

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

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

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

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

which is linear separately in each ${\displaystyle \mathbf {u} ^{(i)}}$  (that is, ${\displaystyle F}$  is multilinear), symmetric in the ${\displaystyle \mathbf {u} ^{(i)},}$  and such that
${\displaystyle F\left(\mathbf {u} ,\mathbf {u} ,\ldots ,\mathbf {u} \right)=f(\mathbf {u} ).}$

The polar form of ${\displaystyle f}$  is given by the following construction

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

In other words, ${\displaystyle F}$  is a constant multiple of the coefficient of ${\displaystyle \lambda _{1}\lambda _{2}\ldots \lambda _{d}}$  in the expansion of ${\displaystyle f\left(\lambda _{1}\mathbf {u} ^{(1)}+\cdots +\lambda _{d}\mathbf {u} ^{(d)}\right).}$

Examples

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

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

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

More generally, if ${\displaystyle f}$  is any quadratic form then the polarization of ${\displaystyle f}$  agrees with the conclusion of the polarization identity.

A cubic example. Let ${\displaystyle f(x,y)=x^{3}+2xy^{2}.}$  Then the polarization of ${\displaystyle f}$  is given by

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

Mathematical details and consequences

The polarization of a homogeneous polynomial of degree ${\displaystyle d}$  is valid over any commutative ring in which ${\displaystyle d!}$  is a unit. In particular, it holds over any field of characteristic zero or whose characteristic is strictly greater than ${\displaystyle d.}$

The polarization isomorphism (by degree)

For simplicity, let ${\displaystyle k}$  be a field of characteristic zero and let ${\displaystyle A=k[\mathbf {x} ]}$  be the polynomial ring in ${\displaystyle n}$  variables over ${\displaystyle k.}$  Then ${\displaystyle A}$  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
${\displaystyle A_{d}\cong \operatorname {Sym} ^{d}k^{n}}$

where ${\displaystyle \operatorname {Sym} ^{d}}$  is the ${\displaystyle d}$ -th symmetric power of the ${\displaystyle n}$ -dimensional space ${\displaystyle k^{n}.}$

These isomorphisms can be expressed independently of a basis as follows. If ${\displaystyle V}$  is a finite-dimensional vector space and ${\displaystyle A}$  is the ring of ${\displaystyle k}$ -valued polynomial functions on ${\displaystyle V}$  graded by homogeneous degree, then polarization yields an isomorphism

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

The algebraic isomorphism

Furthermore, the polarization is compatible with the algebraic structure on ${\displaystyle A,}$ so that

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

where ${\displaystyle \operatorname {Sym} ^{\cdot }V^{*}}$  is the full symmetric algebra over ${\displaystyle V^{*}.}$

Remarks

• For fields of positive characteristic ${\displaystyle p,}$  the foregoing isomorphisms apply if the graded algebras are truncated at degree ${\displaystyle p-1.}$
• There do exist generalizations when ${\displaystyle V}$  is an infinite dimensional topological vector space.