Polarization of an algebraic form

Summary

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 techniqueEdit

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  

ExamplesEdit

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 consequencesEdit

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)Edit

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 isomorphismEdit

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

 
where   is the full symmetric algebra over  

RemarksEdit

  • 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 alsoEdit

ReferencesEdit

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