PlanetMath: weighted homogeneous polynomial

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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

variables and take integers $d_1, d_2, \ldots, d_n$ . The polynomial

is said to be weighted homogeneous of degree if for all

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:

12-00 (Field theory and polynomials :: General reference works )

Pending Errata and Addenda

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