order and degree of polynomial

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

Then the total degree of $f$ is given by:

$$\mathrm{deg}f=sup\{i+j|a_{ij}\ne 0\}$$

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

Similarly the order of $f$ is given by:

$$\mathrm{ord}f=inf\{i+j|a_{ij}\ne 0\}$$

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

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.