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.

Examples

Example 1. Here is a homogeneous polynomial of degree 5, in two variables:

[x^5 + 2 x^3 y^2 + 9 x^1 y^4]

The sum of the indices 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:

[\beginT: & \underbrace & \to & Y\\ & n & &\\\end]

the homogeneous polynomial [\widehat(x)] of degree n associated with T is simply

[\widehat(x) = T(x,x,\ldots,x)]

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

[\widehat(ax) = a^n \widehat(x)]

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

For the case n=2, the tensor is simply a matrix, and such a homogeneous polynomial is known as a quadratic form.

Homogeneous polynomial

Examples

See also