# monomial

A *monomial ^{}* is a product of non-negative powers of variables. It may also include an optional coefficient (which is sometimes ignored when discussing particular properties of monomials). A polynomial

^{}can be thought of as a sum over a set of monomials.

For example, the following are monomials.

$$\begin{array}{ccc}\hfill 1\hfill & \hfill x\hfill & \hfill {x}^{2}y\hfill \\ & & \\ \hfill xyz\hfill & \hfill 3{x}^{4}{y}^{2}{z}^{3}\hfill & \hfill -z\hfill \end{array}$$ |

If there are $n$ variables from which a monomial may be formed, then
a monomial may be represented without its coefficient as a vector of $n$
naturals. Each position in this vector would correspond to a particular
variable, and the value of the element at each position would correspond
to the power of that variable in the monomial. For instance, the monomial
${x}^{2}y{z}^{3}$ formed from the set of variables $\{w,x,y,z\}$
would be represented as ${\left(\begin{array}{cccc}\hfill 0\hfill & \hfill 2\hfill & \hfill 1\hfill & \hfill 3\hfill \end{array}\right)}^{T}$. A constant would be a zero vector^{}.

Given this representation^{}, we may define a few more concepts. First, the
*degree of a monomial* is the sum of the elements of its vector representation. Thus, the degree of ${x}^{2}y{z}^{3}$ is $0+2+1+3=6$,
and the degree of a constant is 0. If a polynomial is represented as a sum
over a set of monomials, then the degree of a polynomial can be defined as the
degree of the monomial of largest degree belonging to that polynomial.

Title | monomial |
---|---|

Canonical name | Monomial |

Date of creation | 2013-03-22 12:34:32 |

Last modified on | 2013-03-22 12:34:32 |

Owner | bbukh (348) |

Last modified by | bbukh (348) |

Numerical id | 5 |

Author | bbukh (348) |

Entry type | Definition |

Classification | msc 12-00 |

Defines | degree of a monomial |