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 be a polynomial in variables Suppose that is homogeneous of degree which means that

Let be a collection of indeterminates with so that there are variables altogether. The polar form of is a polynomial

which is linear separately in each (that is, is multilinear), symmetric in the and such that

The polar form of is given by the following construction

In other words, is a constant multiple of the coefficient of in the expansion of

Examples

A quadratic example. Suppose that and is the quadratic form

Then the polarization of is a function in and given by

More generally, if is any quadratic form then the polarization of agrees with the conclusion of the polarization identity.

A cubic example. Let Then the polarization of is given by

Mathematical details and consequences

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

The polarization isomorphism (by degree)

For simplicity, let be a field of characteristic zero and let be the polynomial ring in variables over Then is graded by degree, so that

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

where is the -th symmetric power of the -dimensional space

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

The algebraic isomorphism

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

where is the full symmetric algebra over

Remarks

  • For fields of positive characteristic the foregoing isomorphisms apply if the graded algebras are truncated at degree
  • There do exist generalizations when is an infinite dimensional topological vector space.

See also

References

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