weighted homogeneous polynomial

Let ${\mathbb{F}}$ be either the real or complex numbers.

Definition.

Let $p\colon{\mathbb{F}}^{n}\to{\mathbb{F}}$ be a polynomial in $n$ variables and take integers $d_{1},d_{2},\ldots,d_{n}$ . The polynomial $p$ is said to be weighted homogeneous of degree $k$ if for all $t>0$ we have

p(t^{d_{1}}x_{1},t^{d_{2}}x_{2},\ldots,t^{d_{n}}x_{n})=t^{k}p(x_{1},x_{2},% \ldots,x_{n}).

The $d_{1},\ldots,d_{n}$ are called the weights of the variables $x_{1},\ldots,x_{n}$ .

Note that if $d_{1}=d_{2}=\ldots=d_{n}=1$ then this definition is the standard homogeneous polynomial.

Title	weighted homogeneous polynomial
Canonical name	WeightedHomogeneousPolynomial
Date of creation	2013-03-22 15:21:18
Last modified on	2013-03-22 15:21:18
Owner	jirka (4157)
Last modified by	jirka (4157)
Numerical id	5
Author	jirka (4157)
Entry type	Definition
Classification	msc 12-00