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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, in particular in
algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
, 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 ...
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 a mo ...
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 which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
,
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 descr ...
, and
representation theory Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...
. 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 mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
in n variables \mathbf = \left(u_1, u_2, \ldots, u_n\right). Suppose that f is homogeneous of degree d, which means that indeterminates 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 t ...
f(\mathbf) = x^2 + 3 x y + 2 y^2. Then the polarization of f is a function in \mathbf^ = (x^, y^) and \mathbf^ = (x^, y^) 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 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 In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a pr ...
ring (The) Ring(s) may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell Arts, entertainment, and media Film and TV * ''The Ring'' (franchise), a ...
in which d! is a
unit Unit may refer to: General measurement * Unit of measurement, a definite magnitude of a physical quantity, defined and adopted by convention or by law **International System of Units (SI), modern form of the metric system **English units, histo ...
. In particular, it holds over any field 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 mathbf/math> be the
polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, ...
in n variables over k. Then A is graded by degree, so that A = \bigoplus_d A_d. The polarization of algebraic forms then induces an
isomorphism In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
of
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
s 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 th ...
. These isomorphisms can be expressed independently of a basis as follows. If V is a
finite-dimensional In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, ยง2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to d ...
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^ 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 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