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Terms From Foreign Languages Used In Mathematics Number Variable Basic Polynomial Power Series

1	Basic Polynomial
1	Terms From Foreign Languages Used In Mathematics
1	Variable
1	Number
3	Power Series

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Order And Degree Of Polynomial

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.