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.
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[edit] Examples
Example 1. Here is a homogeneous polynomial of degree 5, in two variables:
- <math>x^5 + 2 x^3 y^2 + 9 x^1 y^4</math>
The sum of the exponents in each term is always 5.
Example 2. A homogeneous polynomial may be constructed from a tensor of order n. Thus, if X is a vector space, and Y is another space, then, given a tensor T:
- <math>
\begin{matrix} T: & \underbrace{X \times X \times \cdots \times X} & \to & Y\\
& n & &\\
\end{matrix} </math>
the homogeneous polynomial <math>\widehat{T}(x)</math> of degree n associated with T is simply
- <math>\widehat{T}(x) = T(x,x,\ldots,x)</math>
In this form, it is clear that a homogeneous polynomial is a homogeneous function of degree n. That is, for a scalar a, one has
- <math>\widehat{T}(ax) = a^n \widehat{T}(x)</math>
which follows immediately from the multi-linearity of the tensor.
[edit] Quadratic forms
For the case n= 2, the tensor is simply a matrix. A homogeneous polynomial is a quadratic form. The theory of quadratic forms is very extensive and has numerous applications all over mathematics and theoretical physics.
[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] History
In the mathematics of the nineteenth century, an important role was played by algebraic forms.
The two obvious areas where these would be applied were projective geometry, and number theory (less then 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.
