# zero polynomial

The zero polynomial^{} in a ring $R[X]$ of polynomials^{} over a ring $R$ is the identity element^{} 0 of this polynomial ring:

$$f+\text{\U0001d7ce}=\text{\U0001d7ce}+f=f\mathit{\hspace{1em}}\forall f\in R[X]$$ |

So the zero polynomial is also the absorbing element for the multiplication of polynomials.

All coefficients of the zero polynomial are equal to 0, i.e.

$$\text{\U0001d7ce}:=(0,\mathrm{\hspace{0.17em}0},\mathrm{\hspace{0.17em}0},\mathrm{\dots}).$$ |

Because always

$$f\cdot \text{\U0001d7ce}=\text{\U0001d7ce}$$ |

and because in general $\mathrm{deg}(fg)=\mathrm{deg}(f)+\mathrm{deg}(g)$ when $R$ has no zero divisors^{}, one may define that that the zero polynomial has no degree (http://planetmath.org/Polynomial) at all, or alternatively that

$$\mathrm{deg}(\text{\U0001d7ce})=-\mathrm{\infty}$$ |

(see the extended real numbers).

Title | zero polynomial |
---|---|

Canonical name | ZeroPolynomial |

Date of creation | 2013-03-22 14:46:58 |

Last modified on | 2013-03-22 14:46:58 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 13 |

Author | pahio (2872) |

Entry type | Definition |

Classification | msc 13P05 |

Classification | msc 11C08 |

Classification | msc 12E05 |

Related topic | PolynomialRingOverIntegralDomain |

Related topic | OrderAndDegreeOfPolynomial |

Related topic | MinimalPolynomialEndomorphism |