Homogeneous polynomial

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For other meanings, see homogeneous (mathematics)

In mathematics, a homogeneous polynomial is a polynomial whose terms are monomials all having the same total degree; or are elements of the same dimension. For example, x5 + 2x3y2 + 9x1y4 is a homogeneous polynomial of degree 5, in two variables; the sum of the exponents in each term is always 5. An algebraic form, or simply form, is another name for a homogeneous polynomial. A homogeneous polynomial of degree 2 is a quadratic form, and may be simply represented as a symmetric matrix. The theory of algebraic forms is very extensive, and has numerous applications all over mathematics and theoretical physics.

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[edit] Symmetric tensors

Homogeneous polynomials over a vector space may be constructed directly from symmetric tensors, and vice versa. For vector spaces over the real or complex numbers, the set of homogeneous polynomials and symmetric tensors are in fact isomorphic. This relationship is often expressed as follows.

Let X and Y be vector spaces, and let T the multi-linear map or symmetric tensor

\begin{matrix} T: & \underbrace{X \times X \times \cdots \times X} & \to & Y\\ & n & &\\ \end{matrix}

Define the diagonal operator Δ as

\begin{matrix} \Delta: & X &\to &X^n \\ & x &\mapsto &(x,x,\dots,x) \\ \end{matrix}

The homogeneous polynomial \widehat{T} of degree n associated with T is simply \widehat{T} = T \circ \Delta, so that

\widehat{T}(x) = (T \circ \Delta) (x) = T(x,x,\ldots,x)

Written this way, it is clear that a homogeneous polynomial is a homogeneous function of degree n. That is, for a scalar a, one has

\widehat{T}(ax) = a^n \widehat{T}(x)

which follows immediately from the multi-linearity of the tensor.

Conversely, given a homogeneous polynomial P, one may construct the corresponding symmetric tensor \check{P} by means of the polarization formula:

\check{P}(x_1,x_2,\cdots x_n) = \frac{1}{2^n n!} \sum_{\varepsilon_i=\pm 1 \atop 1\le i\le n} \varepsilon_1\varepsilon_2\cdots\varepsilon_n P\left(\sum_{i=1} \varepsilon_i x_i\right)

Let \mathcal{L}(X^n,Y) denote the space of symmetric tensors of rank n, and let \mathcal{P}(X,Y) denote the space of homogeneous polynomials of degree n. If the vector spaces X and Y are over the reals or the complex numbers (or more generally, over a field of characteristic zero), then these two spaces are isomorphic, with the mappings given by hat and check:

\widehat{\;}: \mathcal{L}(X^n,Y) \to \mathcal{P}(X,Y)

and

\check{\;}: \mathcal{P}(X,Y) \to \mathcal{L}(X^n,Y)

[edit] Algebraic forms in general

Algebraic form, or simply form, is another term for homogeneous polynomial. These then generalise from quadratic forms to degrees 3 and more, and have in the past also been known as quantics. To specify a type of form, one has to give its degree of a form, and number of variables n. A form is over some given field K, if it maps from Kn to K, where n is the number of variables of the form.

A form over some field K in n variables represents 0 if there exists an element

(x1,...,xn)

in Kn such that at least one of the

xi (i=1,...,n)

is not equal to zero.

[edit] Basic properties

The number of different homogeneous monomials of degree M in N variables is \frac{(M+N-1)!}{M!(N-1)!}

The Taylor series for a homogeneous polynomial P expanded at point x may be written as

\begin{matrix} P(x+y)= \sum_{j=0}^n {n \choose j} \check{P} ( &\underbrace{x,x,\dots ,x} & \underbrace{y,y,\dots ,y} ). \\ & j & n-j\\ \end{matrix}

Another useful identity is

\begin{matrix} P(x)-P(y)= \sum_{j=0}^{n-1} {n \choose j} \check{P} ( &\underbrace{y,y,\dots ,y} & \underbrace{(x-y),(x-y),\dots ,(x-y)} ). \\ & j & n-j\\ \end{matrix}

[edit] History

Algebraic forms played an important role in nineteenth century mathematics.

The two obvious areas where these would be applied were projective geometry, and number theory (then less in fashion). The geometric use was connected with invariant theory. There is a general linear group acting on any given space of quantics, and this group action is potentially a fruitful way to classify certain algebraic varieties (for example cubic hypersurfaces in a given number of variables).

In more modern language the spaces of quantics are identified with the symmetric tensors of a given degree constructed from the tensor powers of a vector space V of dimension m. (This is straightforward provided we work over a field of characteristic zero). That is, we take the n-fold tensor product of V with itself and take the subspace invariant under the symmetric group as it permutes factors. This definition specifies how GL(V) will act.

It would be a possible direct method in algebraic geometry, to study the orbits of this action. More precisely the orbits for the action on the projective space formed from the vector space of symmetric tensors. The construction of invariants would be the theory of the co-ordinate ring of the 'space' of orbits, assuming that 'space' exists. No direct answer to that was given, until the geometric invariant theory of David Mumford; so the invariants of quantics were studied directly. Heroic calculations were performed, in an era leading up to the work of David Hilbert on the qualitative theory.

For algebraic forms with integer coefficients, generalisations of the classical results on quadratic forms to forms of higher degree motivated much investigation.

[edit] See also