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# 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.

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