PlanetMath: order and degree of polynomial

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (211)
Orphanage
Unclass'd (1)
Unproven (472)
Corrections (36)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

order and degree of polynomial

(Definition)

Let be a polynomial in two variables, viz. $f(x,y) = \sum_{i,j} a_{ij}x^i y^j$ .¹

Then the total degree of is given by:

$\begin{displaymath} \mathrm{deg} f = \sup\{i+j \vert a_{ij} \neq 0\} \end{displaymath}$

Note the degree of the zero-polynomial is $-\infty$ , since $\sup\emptyset$ (per definition) is $-\infty$ , thus $\mathrm{deg} f \in \mathbb{N}\cup\{0\}\cup\{-\infty\}$ .

Similarly the order of is given by:

$\begin{displaymath} \mathrm{ord} f = \inf\{i+j \vert a_{ij} \neq 0\} \end{displaymath}$

Note the order of the zero-polynomial is $\infty$ (because $\inf\emptyset = \infty$ ). Thus $\mathrm{ord} f \in \mathbb{N}\cup\{0\}\cup\{\infty\}$ .

Please note that the term order is not as common as degree. In fact, it is perhaps more frequently associated with power series (a form of generalized polynomials) than with ordinary polynomials. Also be aware that the term order occasionally is used as a synonym for degree.

Footnotes

....¹: In order to simplify the notation, the definition is given in terms of a polynomial in two variables, however the definition naturally scales to any number of variables.

"order and degree of polynomial" is owned by jgade.

(view preamble)