Quasi-homogeneous polynomial

In algebra, a multivariate polynomial f ( x ) = ∑ α a α x α , where α = ( i 1 , … , i r ) ∈ N r , and x α = x 1 i 1 ⋯ x r i r , {\displaystyle f(x)=\sum _{\alpha }a_{\alpha }x^{\alpha }{\text{, where }}\alpha =(i_{1},\dots ,i_{r})\in \mathbb {N} ^{r}{\text{, and }}x^{\alpha }=x_{1}^{i_{1}}\cdots x_{r}^{i_{r}},} is quasi-homogeneous or weighted homogeneous, if there exist r integers w 1 , … , w r {\displaystyle w_{1},\ldots ,w_{r}} , called weights of the variables, such that the sum w = w 1 i 1 + ⋯ + w r i r {\displaystyle w=w_{1}i_{1}+\cdots +w_{r}i_{r}} is the same for all nonzero terms of f. This sum w is the weight or the degree of the polynomial.

Source: Wikipedia — Quasi-homogeneous polynomial (CC BY-SA 4.0)

Quasi-homogeneous polynomial

In algebra, a multivariate polynomial f ( x ) = ∑ α a α x α , where α = ( i 1 , … , i r ) ∈ N r , and x α = x 1 i 1 ⋯ x r i r , {\displaystyle f(x)=\sum _{\alpha }a_{\alpha }x^{\alpha }{\text{, where }}\alpha =(i_{1},\dots ,i_{r})\in \mathbb {N} ^{r}{\text{, and }}x^{\alpha }=x_{1}^{i_{1}}\cdots x_{r}^{i_{r}},} is quasi-homogeneous or weighted homogeneous, if there exist r integers w 1 , … , w r {\displaystyle w_{1},\ldots ,w_{r}} , called weights of the variables, such that the sum w = w 1 i 1 + ⋯ + w r i r {\displaystyle w=w_{1}i_{1}+\cdots +w_{r}i_{r}} is the same for all nonzero terms of f. This sum w is the weight or the degree of the polynomial.

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Source: Wikipedia "Quasi-homogeneous polynomial" · CC BY-SA 4.0

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