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weighted homogeneous polynomial (Definition)

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

Definition 1   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
$\displaystyle p(t^{d_1} x_1,t^{d_2} x_2,\ldots,t^{d_n} x_n) = t^k p(x_1,x_2,\ldots,x_n) .$    

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.



"weighted homogeneous polynomial" is owned by jirka.
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Cross-references: homogeneous polynomial, weights, homogeneous of degree, integers, variables, polynomial, complex numbers, real
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This is version 2 of weighted homogeneous polynomial, born on 2005-06-20, modified 2005-06-21.
Object id is 7177, canonical name is WeightedHomogeneousPolynomial.
Accessed 977 times total.

Classification:
AMS MSC12-00 (Field theory and polynomials :: General reference works )

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