Polarization of an algebraic form

This article is about formulas for higher-degree polynomials. For formula that relates norms to inner products, see Polarization identity.

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

Examples

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 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 $A=k[\mathbf {x} ]$ 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 isomorphism

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

$$A\cong \operatorname {Sym} ^{\bullet }V^{*}$$

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

Remarks

• 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.

See also

• Homogeneous function

References

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