weighted homogeneous polynomial

Let $𝔽$ be either the real or complex numbers.

Definition.

Let $p:{𝔽}^n\to 𝔽$ be a polynomial in $n$ variablesand take integers $d_1,d_2,\mathrm{\dots },d_n$.The polynomial $p$ is said to beweighted homogeneous of degree $k$ if for all $t>0$ we have

$$p(t^{d_1}{x}_1,t^{d_2}{x}_2,\mathrm{\dots },t^{d_n}{x}_n)=t^{k}p(x_1,x_2,\mathrm{\dots },x_n).$$

The $d_1,\mathrm{\dots },d_n$ are called the weights of the variables $x_1,\mathrm{\dots },x_n$.

Note that if $d_1=d_2=\mathrm{\dots }=d_n=1$ then this definition is the standard homogeneous polynomial.