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

be a polynomial is homogeneous of degree

be a collection of indeterminates variables altogether. The polar form

Examples

A quadratic example. is the quadratic form agrees with the conclusion of the polarization identity.

A cubic example.

Mathematical details and consequences

is valid over any commutative ring is a unit. In particular, it holds over any field of characteristic

The polarization isomorphism (by degree)

be the polynomial ring is graded The polarization of algebraic forms then induces an isomorphism of vector spaces-th symmetric power.

These isomorphisms can be expressed independently of a basis is a finite-dimensional

The algebraic isomorphism

is the full symmetric algebra

Remarks

• is an infinite-dimensional topological vector space.

See also

References

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