In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, in particular in algebra

Algebra

Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...

, polarization is a technique for expressing a homogeneous polynomial

Homogeneous polynomial

In mathematics, a homogeneous polynomial is a polynomial whose monomials with nonzero coefficients all have thesame total degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is 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

Algebraic geometry

Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

, invariant theory

Invariant theory

Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties from the point of view of their effect on functions...

, and representation theory

Representation theory

Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...

. Polarization and related techniques form the foundations for Weyl's invariant theory.

The technique

The fundamental ideas are as follows. Let f(u) be a polynomial in n variables u = (u₁, u₂, ..., 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 indeterminate

Indeterminate

Indeterminate has a variety of meanings in mathematics:* Indeterminate * Indeterminate system* Indeterminate equation* Statically indeterminate* Indeterminate formIt is also a term in botany and gardening:*Indeterminate growth...

s with u⁽ⁱ⁾ = (u₁⁽ⁱ⁾, u₂⁽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 λ₁ λ₂...λ_d in the expansion of f(λ₁u⁽¹⁾ + ... + λ_du^(d)).

Examples

• Suppose that x=(x,y) and f(x) is the quadratic form
  Quadratic form
  In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....
  
  

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
  Polarization identity
  In mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space. Let \|x\| \, denote the norm of vector x and \langle x, \ y \rangle \, the inner product of vectors x and y...
  
  .

• A cubic example. Let f(x,y)=x³ + 2xy². Then the polarization of f is given by

Mathematical details and consequences

The polarization of a homogeneous polynomial of degree d is valid over any commutative ring

Commutative ring

In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....

 in which d! is a unit. In particular, it holds over any field

Field (mathematics)

In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

 of characteristic zero or whose characteristic is strictly greater than d.

The polarization isomorphism (by degree)

For simplicity, let k be a field of characteristic zero and let A=k[x] be the polynomial ring

Polynomial ring

In mathematics, especially in the field of abstract algebra, a polynomial ring is a ring formed from the set of polynomials in one or more variables with coefficients in another ring. Polynomial rings have influenced much of mathematics, from the Hilbert basis theorem, to the construction of...

 in n variables over k. Then A is graded by degree

Degree of a polynomial

The degree of a polynomial represents the highest degree of a polynominal's terms , should the polynomial be expressed in canonical form . The degree of an individual term is the sum of the exponents acting on the term's variables...

, 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 kⁿ.

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

The algebraic isomorphism

Furthermore, the polarization is compatible with the algebraic structure on A, so that
where Sym^.V^* is the full symmetric algebra

Symmetric algebra

In mathematics, the symmetric algebra S on a vector space V over a field K is the free commutative unital associative algebra over K containing V....

 over V^*.

Remarks

• 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
  Topological vector space
  In mathematics, a topological vector space is one of the basic structures investigated in functional analysis...
  
  .