Polarization of An Algebraic Form

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## The technique

## Examples

## Mathematical details and consequences

### The polarization isomorphism (by degree)

### The algebraic isomorphism

### Remarks

## References

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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 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 fundamental ideas are as follows. Let *f*(**u**) be a polynomial in *n* variables **u** = (*u*_{1}, *u*_{2}, ..., *u*_{n}). Suppose that *f* is homogeneous of degree *d*, which means that

*f*(*t***u**) =*t*^{d}*f*(**u**) for all*t*.

Let **u**^{(1)}, **u**^{(2)}, ..., **u**^{(d)} be a collection of indeterminates with **u**^{(i)} = (*u*_{1}^{(i)}, *u*_{2}^{(i)}, ..., *u*_{n}^{(i)}), so that there are *dn* variables altogether. The **polar form** of *f* is a polynomial

*F*(**u**^{(1)},**u**^{(2)}, ...,**u**^{(d)})

which is linear separately in each **u**^{(i)} (i.e., *F* is multilinear), symmetric in the **u**^{(i)}, and such that

*F*(**u**,**u**, ...,**u**)=*f*(**u**).

The polar form of *f* is given by the following construction

In other words, *F* is a constant multiple of the coefficient of ?_{1} ?_{2}...?_{d} in the expansion of *f*(?_{1}**u**^{(1)} + ... + ?_{d}**u**^{(d)}).

- Suppose that
**x**=(*x*,*y*) and*f*(**x**) is the quadratic form

Then the polarization of *f* is a function in **x**^{(1)} = (*x*^{(1)}, *y*^{(1)}) and **x**^{(2)} = (*x*^{(2)}, *y*^{(2)}) given by

- 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}+ 2*xy*^{2}. Then the polarization of*f*is given by

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*.

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

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

where *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

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

where *Sym*^{⋅}*V*^{*} is the full symmetric algebra over *V*^{*}.

- 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.

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

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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