# weighted homogeneous polynomial

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

###### Definition.

Let $p:{\mathbb{F}}^{n}\to \mathbb{F}$ be a polynomial^{} in $n$ variables
and take integers ${d}_{1},{d}_{2},\mathrm{\dots},{d}_{n}$.
The polynomial $p$ is said to be
weighted homogeneous of degree $k$ if for all $t>0$ we have

$$p({t}^{{d}_{1}}{x}_{1},{t}^{{d}_{2}}{x}_{2},\mathrm{\dots},{t}^{{d}_{n}}{x}_{n})={t}^{k}p({x}_{1},{x}_{2},\mathrm{\dots},{x}_{n}).$$ |

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^{}.

Title | weighted homogeneous polynomial |
---|---|

Canonical name | WeightedHomogeneousPolynomial |

Date of creation | 2013-03-22 15:21:18 |

Last modified on | 2013-03-22 15:21:18 |

Owner | jirka (4157) |

Last modified by | jirka (4157) |

Numerical id | 5 |

Author | jirka (4157) |

Entry type | Definition |

Classification | msc 12-00 |