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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, in particular in
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
, polarization is a technique for expressing a
homogeneous polynomial In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; t ...
in a simpler fashion by adjoining more variables. Specifically, given a homogeneous polynomial, polarization produces a unique symmetric
multilinear form In abstract algebra and multilinear algebra, a multilinear form on a vector space V over a field K is a map :f\colon V^k \to K that is separately ''K''-linear in each of its ''k'' arguments. More generally, one can define multilinear forms on ...
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 is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
,
invariant theory Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit descri ...
, and
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 studies modules over these abstract algebraic structures. In essen ...
. Polarization and related techniques form the foundations for Weyl's invariant theory.


The technique

The fundamental ideas are as follows. Let f(\mathbf) be a polynomial in n variables \mathbf = \left(u_1, u_2, \ldots, u_n\right). Suppose that f is homogeneous of degree d, which means that indeterminate Indeterminate may refer to: In mathematics * Indeterminate (variable), a symbol that is treated as a variable * Indeterminate system, a system of simultaneous equations that has more than one solution * Indeterminate equation, an equation that ha ...
s with \mathbf^ = \left(u^_1, u^_2, \ldots, u^_n\right), so that there are d n variables altogether. The polar form of f is a polynomial F\left(\mathbf, \mathbf, \ldots, \mathbf\right) = f(\mathbf). The polar form of f is given by the following construction quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to a ...
 f(\mathbf) = x^2 + 3 x y + 2 y^2. Then the polarization of f is a function in \mathbf^ = \left(x^, y^\right) and \mathbf^ = \left(x^, y^\right) given by F\left(\mathbf^, \mathbf^\right) = x^ x^ + \frac x^ y^ + \frac x^ y^ + 2 y^ y^. More generally, if f is any quadratic form then the polarization of f agrees with the conclusion of the
polarization identity In linear algebra, a branch of 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. If a norm arises from an inner product then t ...
. A cubic example. Let f(x, y) = x^3 + 2xy^2. Then the polarization of f is given by F\left(x^, y^, x^, y^, x^, y^\right) = x^ x^ x^ + \frac x^ y^ y^ + \frac x^ y^ y^ + \frac x^ y^ y^.


Mathematical details and consequences

The polarization of a homogeneous polynomial of degree d is valid over any
commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...
in which d! is a unit. In particular, it holds over any
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
of
characteristic zero In mathematics, the characteristic of a ring , often denoted , is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive id ...
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 mathbf/math> be the
polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) ...
in n variables over k. Then A is graded by
degree Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathematics ...
, so that A = \bigoplus_d A_d. The polarization of algebraic forms then induces an isomorphism of vector spaces in each degree A_d \cong \operatorname^d k^n where \operatorname^d is the d-th
symmetric power In mathematics, the ''n''-th symmetric power of an object ''X'' is the quotient of the ''n''-fold product X^n:=X \times \cdots \times X by the permutation action of the symmetric group \mathfrak_n. More precisely, the notion exists at least in the ...
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 A_d \cong \operatorname^d V^*.


The algebraic isomorphism

Furthermore, the polarization is compatible with the algebraic structure on A,so that A \cong \operatorname^\cdot V^* where \operatorname^ V^* is the full
symmetric algebra In mathematics, the symmetric algebra (also denoted on a vector space over a field is a commutative algebra over that contains , and is, in some sense, minimal for this property. Here, "minimal" means that satisfies the following universal ...
over V^*.


Remarks

* For fields of
positive characteristic In mathematics, the characteristic of a ring , often denoted , is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive id ...
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 In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
.


See also

*


References

*
Claudio Procesi Claudio Procesi (born 31 March 1941 in Rome) is an Italian mathematician, known for works in algebra and representation theory. Career Procesi studied at the Sapienza University of Rome, where he received his degree (Laurea) in 1963. In 1966 he ...
(2007) ''Lie Groups: an approach through invariants and representations'', Springer, . {{DEFAULTSORT:Polarization Of An Algebraic Form Abstract algebra Homogeneous polynomials