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'order and degree of polynomial'
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| Title of object: |
order and degree of polynomial |
| Canonical Name: |
OrderAndDegreeOfPolynomial |
| Type: |
Definition |
| Created on: |
2003-05-20 19:42:43.334319-04 |
| Modified on: |
2003-05-20 19:49:34.06332-04 |
| Classification: |
msc:12-00 |
Revision comment (for changes between this and next version):
| Changes for correction #1919 ('"order" and "degree" defines'). |
Preamble:
\usepackage{amssymb}
\usepackage{amsfonts} |
Content:
Let $f$ be a polynomial in two variables, viz. $f(x,y) = \sum_{i,j} a_{ij}x^i y^j$. Then the \emph{degree} of $f$ is given by:
\mathrm{deg} f = \sup\{i+j | a_{ij} \neq 0\}
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 \emph{order} of $f$ is given by:
\mathrm{ord} f = \inf\{i+j | a_{ij} \neq 0\}
Note the order of the zero-polynomial is $\infty$ (because $\inf\emptyset = \infty$). Thus $\mathrm{ord} f \in \mathbb{N}\cup\{0\}\cup\{\infty\}$.
Note that 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. |
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