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
- Homogeneous function – Function with a multiplicative scaling behaviour
References
- Claudio Procesi (2007) Lie Groups: an approach through invariants and representations, Springer, ISBN 9780387260402 .