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 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

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

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

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

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

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 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

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

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

The algebraic isomorphism

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

where ${\displaystyle \operatorname {Sym} ^{\bullet }V^{*}}$ 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.

See also

• Homogeneous function – Function with a multiplicative scaling behaviour

References

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