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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.
For example, if is a compact Kähler manifold, = (,) is the -th cohomology group of X with integer coefficients, then = (,) is its -th cohomology group with complex coefficients and Hodge theory provides the decomposition of into a direct sum as above, so that these data define a pure Hodge structure of weight .
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 the polarization identity can be used to express this inner product entirely in terms of the norm. The ...
Polarization of an Abelian variety, in the mathematics of complex manifolds; Polarization of an algebraic form, a technique for expressing a homogeneous polynomial in a simpler fashion by adjoining more variables; Polarization identity, expresses an inner product in terms of its associated norm; Polarization (Lie algebra)
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a smooth projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions. Abelian varieties are at the same time among the most studied objects in algebraic geometry ...
Over the complex numbers, the Jacobian variety can be realized as the quotient space V/L, where V is the dual of the vector space of all global holomorphic differentials on C and L is the lattice of all elements of V of the form []: where γ is a closed path in C. In other words,
The associated bilinear form of a quadratic form q is defined by (,) = ((+) ()) = =. Thus, b q is a symmetric bilinear form over K with matrix A . Conversely, any symmetric bilinear form b defines a quadratic form q ( x ) = b ( x , x ) , {\displaystyle q(x)=b(x,x),} and these two processes are the inverses of each other.
The elliptic curve E : 4Y 2 Z = X 3 − XZ 2 in blue, and its polar curve (E) : 4Y 2 = 2.7X 2 − 2XZ − 0.9Z 2 for the point Q = (0.9, 0) in red. The black lines show the tangents to E at the intersection points of E and its first polar with respect to Q meeting at Q.