# homogeneous polynomial

A polynomial^{} $P({x}_{1},\mathrm{\cdots},{x}_{n})$ of degree $k$ is called homogeneous^{} if
$P(c{x}_{1},\mathrm{\cdots},c{x}_{n})={c}^{k}P({x}_{1},\mathrm{\cdots},{x}_{n})$ for all constants $c$.

An equivalent^{} definition is that all terms of the polynomial have the same degree (i.e. $k$).

Observe that a polynomial $P$ is homogeneous iff $\mathrm{deg}P=\mathrm{ord}P$.

As an important example of homogeneous polynomials^{} one can mention the symmetric polynomials^{}.

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

Canonical name | HomogeneousPolynomial |

Date of creation | 2013-03-22 13:21:11 |

Last modified on | 2013-03-22 13:21:11 |

Owner | jgade (861) |

Last modified by | jgade (861) |

Numerical id | 11 |

Author | jgade (861) |

Entry type | Definition |

Classification | msc 12-00 |